Firstly, it must be understood that evolution is not a past occurrence, but an ongoing process that is a universal facet of biology. Everything evolves, by its very nature of being alive. Evolution is part of the process of adaptation, which is one of the basic characteristics of life. That much said, Homo sapiens did not evolve from apes. Homo sapiens is one of the twenty-four extant species of apes. Monkeys and apes did evolve from a common ancestor, but neither apes nor monkeys are descended from one another. The relation of apes (including humans) to monkeys is akin to first cousins: a person is not descended from one's cousin (hopefully), but shares a common ancestor (grandparents) with them.
Note that there is a discussion in the field of cladistics about
whether the ancestor common to both the New World monkeys and the
Old World monkeys could not be called 'monkey' as well. If one
applies strict monophyletic labelling then apes are monkeys since
they evolved from monkeys, and humans, being apes, are also
Humans share a common ancestor with other apes and monkeys, they're not direct descendants of them. Nonhuman apes and monkeys are more like our cousins than our grandparents.
The ancestors of all primates are extinct species. This didn't have to be the case, as one could have survived until now, but it just so happens that none of the currently living primates were among that early group. That early group diverged in a period between 58 mya (million years ago) and 40 mya. There are three major clades (groups of organisms that share a single ancestor) to be concerned with at this point. One became the New World monkeys, living on the American continents. Another was to become the Old World primates, including monkeys and apes throughout Africa and Europe. A third would become various species of monkeys that lived in Asia, but those are all extinct for various reasons. Note that all humans belong to that second group, wherever they might have migrated since.
Later, those Old World primates further developed new species (which doesn't automatically make the older species 'obsolete' or 'doomed' in any way) and created more clades, including apes and monkeys. Later, the ape lineage itself split into two clades: the "lesser ape", the current descendants of which are called gibbons, and the "great ape", which includes all other apes like gorillas, chimps and humans.
The closest living relatives to humans living today are the chimps, of which there are two species: Pan troglodytes (the common chimp) and Pan paniscus (the bonobo chimp). I'll use this divergence as a good spot to explain how the split might happen.
Let's look at the common ancestor of humans and chimps. At some point, some members of this species got separated for whatever reasons from the others for a very long period of time (hundreds of thousands of years, at least). One species continued as it did or adapted to changes, depending on what happened to where they lived. The other adapted in the new place where they lived. Again, it's not completely necessary that this happen. It's entirely possible, though unlikely, that each group would remain unchanged until years later, when they meet back up and begin sharing genes again as they mate. But it didn't happen that way and the two groups diverged genetically until they were two separate species. This continued among the chimps at least until we have the two species today, though there could have been others that didn't survive until now, either because they died off or they reintegrated back into one of the other species at some point.
This happened to humans, as well. An interesting factoid to know is that a small percentage of DNA in all current humans who are not directly of recent African descent comes from the Neanderthal.
As an exercise in logic, You postulate that we evolve and ask how what we evolve from could still be around. This is the same argument as asking "If skyscrapers developed from huts, then why are there still huts?"
One possible answer to that is that huts still have their place.