The answer is, perhaps disappointingly, "not that different to how it is now". Indeed, of all the settled continents, it's probably Africa that has changed the least since the Ice Ages. You might think that this has something to do with Africa being close to the equator; in particular, that it's too close for whopping great sheets of ice to have rolled across the countryside.
Which they didn't, so that much is true. Indeed, the southern hemisphere in general had far less ice cover than Europe, Asia, and North America. That's due mainly to the way that the continents happen to be arranged, with glacial ice sheets only being able to get as far as southern South America (there are fjords in places like Tierra del Fuego). Presumably, sea ice extended much further across the Southern Ocean than it does now, but that would have little effect on land-based animals.
But that's not to say that Africa, or Australia, were unaffected by the Ice Ages. Africa was colder than it is today, and, to begin with at least, rather drier, too. The Sahara and Kalahari deserts were more extensive than they are today, with the semi-desert belt of the Sahel being quite a way south of its present position, running through what are now fairly lush countries such as Guinea, Nigeria, and northern Kenya.
It was the real tropical zones that took the greatest brunt, though. Most of what is now the Congo was, at the height of the glaciations, relatively open savannah, with the rainforests restricted to smaller, fragmented patches. In comparison, while they also shrank considerably, the Amazon and the jungles of Southeast Asia got off much more lightly - the latter in part because lower sea levels meant Indonesia was quite a bit larger.
Africa had no such benefit, because the seabed falls away sufficiently rapidly around it that its coastline hardly changed when the seawaters became locked up in the ice sheets further north. One of the few significant changes was that the Bab-el-Mandeb, through which the Red Sea connects to the Indian Ocean, was at times high and dry, leaving the former as an inland sea. In fact, it may be over this land bridge that modern humans first entered Asia, rather than by the more northerly route through the Sinai.
All of these climatic changes had a dramatic effect on the local wildlife. The early Pleistocene, when the Ice Ages were first gearing up, did see a swathe of extinctions, as animals adapted to the older, warmer, climate died out. Three-toed horses, sabretooth cats, and running hyenas all went extinct in Africa in roughly the first half of the epoch, and many of them right at its beginning. What was different in Africa, though, was what happened when the Ice Ages finished.
On every other human-inhabited continent, the end of the Pleistocene saw a sudden wave of extinction, as animals like woolly mammoths and dire wolves went to their graves. Not so in Africa. Perhaps something like ten species of large mammal went extinct at about this time, but, compared with anywhere else, that's nothing.
Which means, of course, that the animals that came in at the beginning of the Pleistocene, the ones replacing the older forms going extinct then, are pretty much the ones we still see in Africa today. The first zebras, for instance, seem to have entered the continent around 2.5 million years ago, just as the Pleistocene got underway. They were followed by white rhinos, giraffes, elephants, lions, leopards, cheetahs, jackals, and modern hyenas. In many cases, especially among the carnivores, these are the exact same species we have now. Lions, for example, are at least a million years old, and possibly quite a bit more (depending, as always, on where you choose to draw the line).
While a modern time traveller to the early Pleistocene would spot the differences quite quickly, it might take her quite a bit longer to twig what had happened if she arrived in the mid to late parts of the epoch. In the former case, there would be a number of animals more associated with the preceding epoch, the Pliocene, still around. For instance, the presence of sabretooth cats feeding on zebras would be one give-away.
There were also rather a number of pigs in Africa during the Pliocene, and a few of these struggled on into the Pleistocene, before becoming limited to the four species we have today (not counting wild boar, which are generally introduced from Europe and Asia). One of these was the giant warthog (Metridiochoerus andrewsi). Standing around a metre tall at the shoulder, this was about a third again the size of living warthogs, and its huge curving tusks were even more impressive. More interesting, perhaps, were its cheek teeth, which included a single, long, molar with multiple parallel ridges in each jaw. They looked somewhat like the teeth of elephants, a feature that suggests that this animal was primarily a grazer, eating large quantities of grass.
But that's not to say our time traveller wouldn't find clues to an arrival in the late Pleistocene. Because, while there weren't many extinctions at the end of the epoch, there were some.
Among the largest were the long-horned buffaloes (Pelorovis spp.), a group of three or four species that include some of the largest bovines there have ever been. While, admittedly, not that much larger than the living buffalo in body size, their horns were a different matter. These were huge, sweeping, curved affairs, whose bony cores extend over a metre (3 feet) from the head; presumably, when surrounded by the horny material that would have covered them in life, they were even bigger.
Over time, our opinion of how Pelorovis relates to living cattle has changed considerably. They were originally identified as a type of caprine, hence a name that literally translates as "large sheep", but opinion later shifted them to the bovines, and particularly to water buffalo. A more modern interpretation is that Pelorovis may represent two different groups of animal. One of them, represented by P. antiquus, is likely an early African buffalo, despite having very wide horns with upward-pointing tips, that, honestly do look more like those of the water buffaloes of Asia.
The remaining species, of which the better known is B. oldowayensis, had horns that curved backwards, out and down, and then forward - quite a different shape from their supposed relatives. They were also noted for having unusually long and narrow heads, reminiscent of those of wildebeest, and may have grazed in the same manner as those animals. There is some evidence that these animals may have been closely related to, or even the ancestors of, the modern genus Bos, which includes, among others, yak and domestic cattle. This relationship is supported by what appears to be an intermediate form, dating from late Pleistocene Ethiopia.
Of similar height, but with long legs and a slimmer body, was the giant hartebeest (Megalotragus issaci). A type of grazing antelope, related to wildebeest, this was larger and more heavily built than living hartebeest, which are found today across much of the less heavily forested parts of sub-Saharan Africa. It may have been even more specialised for a grazing lifestyle, perhaps because the African grasslands were more extensive then than they are now.
Rather more distinctive perhaps was Sivatherium, a sort of short-necked giraffe. This had a heavily built body, and a thick, muscular, neck, and stood about 1.7 metres (5½ feet) tall at the shoulder. Perhaps its most distinctive feature, though, was the pair of large horn-like structures on its head, differing from true horns in that they were covered with skin and perhaps even hair. These were much more elaborate than the similar structures on the head of giraffes (the technical term is "ossicones"), and, depending on the exact species, were often branched or curved.
Many older reconstructions show sivatheres as looking somewhat like bulky moose with a tapir-like snout. This is probably wrong, and it likely looked more like an okapi, or perhaps an antelope, albeit quite a bit larger and heavier than either of those animals. And, while we can't prove it didn't have a short trunk like a tapir, there's no good reason to suppose that it did, either, so modern pictures tend to leave it out.
These creatures all died out around 12,000 BC, give or take a millennium, just as the Pleistocene was ending. The reasons are likey much the same as for the extinction of their counterparts in the northern hemisphere: a combination of a changing, warmer, climates, and human activity. And yet they are, as I said, among a very small number of African creatures to suffer this fate. They are the odd ones out, and most African wildlife of the Pleistocene is still with us.
One possible reason for this could be that African wildlife, unlike that elsewhere, had already evolved alongside us. Humans entered Australia something like 40,000 years ago, and the Americas not much later than that. Both arrivals had a dramatic effect on wildlife unused to their presence. Europe has been inhabited by modern humans for perhaps twice as long as this, although they shared the continent with Neanderthals for much of that time, and the latter, and their immediate ancestors, had been on the continent since maybe 0.8 million years ago. Even so, only southern Asia has a human presence that comes close to being as long as that of Africa.
How long that is is a matter of definitions. Our subspecies, Homo sapiens sapiens, dates back something like 100,000 years, and there's at least some argument that it might have arisen in southern Asia, rather than Africa. But go back before that time, and you get a blurrier picture as we fade into the species from which we evolved. Where to draw the line, and just how modern does a skull need to look before we consider to belong to our own species? A sufficiently broad definition, for instance, might push us as far back as 0.5 million years ago, and would indisputably place our origin in Africa.
There is considerable debate as to how we divide up the genus Homo, and how different anything has to be before we consider it its own species. We do know that our genus arose, in Africa, round about the beginning of the Pleistocene, so if we take "human" to be a term for that, rather than just our own species, we've been there for over 2 million years. But whether you think our immediate ancestor was H. heidelbergensis, H. rhodesiensis, H. erectus, or H. ergaster, depends as much on how you draw the lines between those species as on what you think happened, and where.
But we can say that, even at the dawn of the Pleistocene, there were tool-using "humans" in East Africa, and they've been there in one form or another, ever since. This was the continent from which we set out to conquer the world.
[Picture by Nobu Tamura, from Wikimedia Commons]