As-salamu Alaykum,
Islam does not deny neither accept the theory of Evolution. In Islam Allah (swt) is known as Al-Bari (the evolver). In the Quran, Allah shows how the earth and the heavens was one and how he has evolved them into the earth and the heavens. Allah shows how he evolved eve from Adam (Adam's rib). Allah also shows how all creatures are created from water (does that mean they were created in the same method?-yes)
We cannot accept that Humans came from monkeys though. Adam was created miraculously in heaven. But one may argue why are there so many similarities among humans and animals. Well as an architect uses the same methods and tools to make different buildings, Allah also used the same methods to make different creatures. Think about it, Earth has a specific set of elements--that means that creation will contain those elements and will be formed from those elements. Just as an architect has a specific set of tools, his method of creation of different buildings will be the same.
Think about it, If God created Adam and had to banish him from heaven, where is he going to put Adam? In mars? No, he will put Adam in a place in which he can live. And this place that Adam lives will require Adam to breathe in Oxygen and other life functions. Well since this Earth allows Adam to breathe oxygen, then why shouldn't the animals there also breathe oxygen. They are both living in the same environment, so they have the same life functions. Now for some people to say that Adam was therefore created from those creatures is absurd. If Adam and those creatures were to exist in the same environment, then they would have close anatomies/physiologies (so they utilize the environments the same way). Even with the evidence of evolution, there are so many missing links. The reason they say this is because it is easy to understand and makes sense. Yes, we know that neanderthals for example once existed. But did you forget that Allah said that he created creatures in Earth and then he wiped them out.
I hope I answered your questions.