Before I begin I would like to make a few things clear.
One: I believe in God, or at least a higher being that created us. To me, it's the only logical answer. I believe it was Leonardo Da Vinci who said "An infinite string of finite events cannot be started by a finite event."
Two: Evolution does not disprove the existence of God. It never has, never will, and can't. It only explains how, not who or why.
Three: The theory of evolution is a theory in the same way as the theory of gravity. It's a scientific theory or "a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment."
Now, on with the show.
When we think of evolution we think of that picture of a fish turning into an ape turning into man. This picture, while accurate, is so incomplete that it tells you nothing about how evolution works. Evolution, while simple, is insanely hard to understand. It is the combination of random chance and inevitability.
Let's start with something simple, the Biston Peppered Moth. Now, this actually happened, we saw this happen. The peppered moth is a light colored moth with the ability to blend into the light colored trees. Standard camouflage, common in nature. When the industrial revolution came around, pollution caused the trees to get dark splotches. This caused the moths to stand out on the trees and start getting picked off by predators. There were other moths of the same family but they were darker. They blended into the dirty trees and thrived.
This is how evolution works on a small scale, or as it's commonly known "Survival of the Fittest". One special trait gives one animal an advantage over another. This could be species wide, a random mutation, or a combination of traits from the parents. This animal ether survives where the other does not, or is determined a better mate then the other. Thus that animal is allowed to mate and pass on it's advantage to it's children.
Now, don't get confused, this is not a directed path, there is no goal when this happens, there is not higher level we're going towards. A bad trait can be passed on as well. Sometimes it results in the death of the animal, thus the trait cannot be passed on. Sometimes the trait survives, this is how we get the oddities we have in the human body. Why do we have a gallbladder if all it seems to do is make us sick? It's just a leftover, and now detrimental, trait from our past.
Now, take this "Survival of the Fittest" and extend it over millions of years. It's completely logical that external appearances can change. This is why humans come in different colors. In an environment with lots of sun and heat, darker skin color helps people live. Where there are lots of clouds, or it's cold so people are covered up a lot and not directly exposed to the sun, the darker skin wasn't required, thus they didn't get that advantage.
It's logical that one group of the same species could evolve in a completely different direction then another group of the same species due to different environments. Given enough time they may not even be able to mate any more. Think of the horse and the zebra. Same kind of animal, they can mate, but their offspring are sterile. It won't be long before they couldn't at all.
Now, I did hear a logical question come up just today. It was followed up by an idiotic example, but the original question was logical. How did they make the switch between asexual reproduction and sexual reproduction. The example was a dog splitting in half to reproduce. This is idiotic due to it being a hypothetical not in any way related to what is being hypothesized. A being that complex would never split in half to reproduce asexually. It would be done like some insects. They can fertilize themselves and give birth to an exact genetic duplicate of themselves. This is how asexual reproduction is done in complex life forms.
To answer the question, you have to look much further back in evolution. Back when multi-celled organisms were really only multi-celled due to it being advantages to be in a group. Strength in numbers. While this change was not genetic, it was still a part of evolution. Back then the organism could divide by splitting itself in half. Just like a group of 20 people could go in two directions by having ten go one way and ten going another. The cells were able to exist independently, they just liked hanging out together.
The concept of internal reproduction came around when cells started specializing. Before, they were all the same and existed in such a simple form that it allowed all cells to touch the environment around them, the water. This is how they gathered nutrients, directly from the water. As these organisms started hanging out in bigger and bigger groups, this turned out not to be possible. That's when cells started to specialize. Some cells started passing nutrients onto the cells beside them, and thus was born a rudimentary digestive system. Threw this same kind of specialization, internal reproduction was born (so to speak).
After that it became advantages to share experiences. That multi-celled organism over there went threw different trials and grew stronger in a different way then this multi-celled organism. Now, they couldn't just tell each other, they weren't sentient. They could only do it by sharing genetic code. And thus sexual reproduction was born.
Now, we've covered the basics (and by basics, I mean the absolute basics) of evolution. There's a lot of chance going on here, lots of chance. What really are the odds that something like this would happen? Million to one, trillion to one, higher? Think of this, space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is, so big that the introduction to the Hitchhiker's Guide to the Galaxy doesn't come anywhere near close enough to describing it. It's so big that mathematically speaking, just by random chance there's an exact clone of you walking around somewhere else in the universe.
There are so many stars in the sky, so many planets around those stars, the question shouldn't be "what are the odds this happened?", a better question would be "what are the odds this wouldn't happen?"
I would like to point out again that all this, all the real study of evolution by people way the hell smarter then me, only answers the question of how (in my opinion, the easiest question to answer). This does not answer the question of who or why. This is where religion comes in.
I know what you're saying, the Bible says God snapped his fingers and suddenly man existed. Think of this, what do you tell your child when he or she asks where babies come from? Do you explain to them the complex concept of sexual reproduction, cell division, and birth? No, you tell them a stork drops it off. The child doesn't have the mental capacity to understand a concept like reproduction. Now think of this, we barely have the mental capacity to understand evolution, we don't have the mental capacity to understand life. Would God waste his/her time explaining the complex path he took to create the universe, the stars, the planets, and cells to someone who just would never be able to understand it? No, God would know better. He would just explain it simply, he snapped his fingers and it happened.