Humans did evolve from apes.
Both sentences are true, and there's an important distinction here. While it's also true that chimpanzees are humanity's closest living relative on the tree of life, the creatures we evolved from were not chimpanzees, but rather a separate, extinct species that gave rise to us both. Yes, these ancient apes were perhaps closer to chimps than to us by most standards of measurement, but that doesn't make them chimps.
These apes gave rise not just to chimps and ourselves, but also to a host of other species in the Homo genus: homo habilus, homo erectus, and homo neanderthalensis. And they were preceded by other species in the Australopithecus genus, of which Lucy is the most famous example.
None of these ancestors of ours were simply replaced by the next in line. Life is not like that. Species fight to stay alive and produce a new generation, but all of these species in our close family tree failed to do so, once. And once is all it takes.
It sounds odd that none of our close relatives survived to the present day. After all, many were more intelligent than the other apes alive today. Some made tools, Neanderthals even had the beginnings of symbolic and spiritual culture. So what happened?
Well, some of these species possibly all changed very gradually over time, as new genes developed from random mutations, and the beneficial ones spread through the population. Some species probably split off into several populations, some staying in the trees, and others spending more time on the ground. Selective pressures altered the populations in different ways, and then at some point one population or the other died out. Perhaps driven to starvation when the other group became more successful at acquiring food.
Actually, it shouldn't seem too surprising that we are the only surviving members of our genus. The vast, vast majority of all species that have existed on earth are extinct. Long ago, even the human species was nearly wiped out. When you compare the gene pool of everyone in the world, it's clear we came from a very small population far back in our past, perhaps a few thousand individuals.
Let's go back to the chimps. Because chimpanzees are probably more similar to our shared relatives than we are, it isn't fair to call them "less evolved." They've survived just as well as we have in these past few million years, with completely different adaptations than our hairless bodies and bulbous brains.
As another example, think of the coelacanth, that fossil fish with four lobed fins that might one day have evolved into legs. Fossils of this fish were known before a living one was discovered by Western science in the 20th century (apparently locals had been eating them for generations), and so the fish was dubbed the "living fossil." But let's give the fish some credit. The oceans have not stayed exactly the same for the millions of years from when its fossilized ancestor swam with the ichthyosaurs to today. To say the coelacanth is unevolved would be unfair and untrue. Adaptations to body chemistry, foraging habits, breeding habits, and possible migration patterns could all have taken place (the continents have been steadily shifting position during this species' time on Earth, for instance) without major changes to the shape of its body.
So this is a toast to all those "under-evolved" species: to the chimpanzees who do not need huge brains to find good food to eat, to the coelacanths who do not need legs to find a mate, to all the amphibians who will reproduce just fine as long as there is freshwater to be found, I say, if it ain't broke, don't fix it.