At the time of this writeup, the Aquatic Ape Hypothesis has been effectively dismantled by the scientific community: most of the 'facts' used to support it are either outright false or else removed from meaningful context in such a way that they appear to contribute to the AAH, even when they support the Savanna Theory more strongly and consistently. Here I shall clarify these falsehoods and failures in academic rigour.
The Mammalian Diving Reflex is present in all mammal species, and all vertebrates have some analogue to it, even non-mammalian species.
All mammals and most vertebrates can voluntarily hold their breath, and nonhuman mammals would regularly die from inhaling their food if this was not the case, since all terrestrial mammals' respiratory tract and digestive tract use the same plumbing; this is also part of why all mammals can get hiccoughs.
Seals and whales do not hold their breath when they dive. The only oxygen they take with them on a dive is what they store in their blood; they actually expel the air from their lungs before a dive. This is why you don't hear about pinnipeds and cetaceans getting the bends. Holding your breath is not uncomfortable due to a lack of oxygen; the oxygen in your lungs is actually quite adequate for a very long span of time, if oxygen is the only concern. The discomfort of holding your breath comes from the carbon dioxide buildup in your blood, raising the pH of your blood and sending a signal to your brain that you need to exhale all of that toxic gas from your lungs. Water pressure makes this sensation even more urgent and uncomfortable. Many deep sea free-divers use lungpacking techniques like pre-dive hyperventilation to raise their blood oxygenation, and they will also exhale instead of inhaling immediately before submerging, but this method is not intuitive or instinctive, and it was arrived-at because of rigorous efforts by free-divers to find an ideal breathing technique. It takes training and practice simply to be able to hyperventilate effectively and without falling unconscious.
All mammal infants are able to float and swim at least minimally in water.
As swimming-capable species go, humans are extremely slow in water and have extremely poor endurance for distance swimming, along with having very inefficient placement of our weight along the length of our body. The same large lungs which are cited as an aquatic advantage are actually a disadvantage to efficient swimming, because they make us "top-buoyant" rather than "top-heavy" in the water. This forces our shoulders to sit much higher in the water than the forward-most body structures on any other aquatic animal, a feature that results in greater drag, lower water displacement per stroke, and significantly increased difficulty maintaining any kind of depth in totally submerged swimming.
Our lungs' size and placement is, however, very useful for long distance endurance running. Our entire physiology is ideally suited for running longer distances without rest than any other animal species can sustain, especially animals historically hunted by humans on land. The closest animals to this kind of endurance are horses and canids - animals which conveniently can also survive in most of the climates where humans can survive; it makes sense that these would be the animals domesticated and assimilated most completely into human life for so much of our species' history. It makes sense that these animals became our partners in hunting.
Most Old World Primates have downward-facing nostrils; humans are hardly unique in this feature. It is still easy to get water in a human nose, and the human nose is structured in such a way that we have a strong olfactory sense compared to other animals with binocular colour vision. Strong senses of smell are useless to mammals underwater, and no aquatic mammal has binocular colour vision. Most aquatic mammals in general have very poor vision; sea otters have much better vision above water than underwater.
All primates have salty tears. Healthy aquatic vision requires greasy tears rather than salty tears, in order to protect the eyes from oceanic saltwater (which has a harmfully different pH from human skin and eyes - something well known to anybody who has ever endured saltwater abscesses).
The "descended" larynx of humans is cited as an unique improvement of our ability to inhale deeply for diving, and part of the AAH bases lesser hypotheses around the apparent lack of descent in other apes' and terrestrial animals' larynges. Even if we are to disregard that deep inhalation is good for running and bad for diving, as was addressed earlier within this writeup, the initial assertion itself is erroneous: chimpanzees, pigs, canids, goats, genus Panthera, gallinaceous gamebirds, passerine birds, and most monkey species all have descended larynges. The other offered 'reason' for descended larynx is that it allows especially low vocalizations which would travel well in water. Low vocalizations travel well in air, too, and a majority of human vocalizations qualify as quite high compared to the vocalizations of elephants, crocodilians, and baleen whales (the three largest groups of animals known to use infrasound to communicate over long distances).
Hairlessness is an effective way to stay cool when running long distances; it is also effective for avoiding fur-inhabiting parasites, reducing the need to spend significant spans of time allopreening. Cranial hair limits sunburns to the top of the head while providing some shade for the eyes. Sea otters have some of the densest fur of any mammal; aquatic life does not automatically mean loss of fur, just as terrestrial life does not automatically mean keeping fur.
Bipedal locomotion makes us significantly taller than animals which have historically predated upon or competed with humans. The benefit of height and binocular long-distance colour vision for a terrestrial, omnivorous animal is obvious: we can see farther than the things that eat us; we can see our prey before it sees us (even-toed ungulates have notoriously poor vision compared to their other senses); we can use projectile weapons with high accuracy. On that last point, humans are among the only primates which can throw objects both far and accurately.
Subcutaneous fat deposits are a logical result of an omnivorous diet containing both fats and carbohydrates, while most other animals can only effectively digest one of those macronutrients effectively. Being able to digest carbs and fats both meant that ancient humans could survive lean hunting seasons on primarily plant diets, then survive seasons of poor vegetation by eating primarily meats and other animal-sourced foods. Hibernatory bears likewise have subcutaneous fat derived from an omnivorous diet, and they are similarly terrestrial animals who can swim but do so relatively inefficiently.
None of these statements wholly eliminate the possibility of aquatic ancestors of modern humans, but if anybody is going to attempt to prove humans are not a purely terrestrial species, they will need to produce much more convincing evidence and arguments than the attempts which have been made so far.
Recommended resources for further exploration of this topic follow:
Bridgeman, B (2003). Psychology & evolution: the origins of mind. SAGE Publications.
Jablonski NG (2008). "Sweat". Skin a natural history. Berkeley: University of California Press.
Langdon JH (1997). "Umbrella hypotheses and parsimony in human evolution: a critique of the Aquatic Ape Hypothesis". Journal of Human Evolution 33 (4): 479–94.
MacLarnon, A.M.; Hewitt, G.P. (1999). "The evolution of human speech: The role of enhanced breathing control". American Journal of Physical Anthropology 109 (3): 341–363.
McHenry HM (2012). "Origin and diversity of early hominid bipedalism". In Reynolds SC; Gallagher A. African Genesis: Perspectives on Hominin Evolution. Cambridge University Press.
Meier, R (2003). The complete idiot's guide to human prehistory. Alpha Books.
Morgan, Elaine (1997). "Chapter 9: The Fat Primate". The Aquatic Ape Hypothesis. Penguin.