The short answer is that humans evolved from apes, and before that apes from monkeys, and so on back to the earliest vertebrates and beyond. There’s a bit of info on the evolution of humans from apes over here, where I answered a slightly related question.
Nowadays we know the sequences of DNA and amino acids for genes and proteins in more and more organisms, and this means you can actually ‘see’ directly the similarities in the DNA – that is, in the instructions and the building blocks of life. The more closely related the species are, the more similar the DNA. For instance, the DNA of humans and chimps, whose last ancestor-in-common probably lived about 5 or 6 million years ago, is more than 98.5% identical.
If the species are more distantly related, so that they last had a common ancestor far longer ago (and we’re talking millions and millions of years), then the similarity in the DNA is less, but still there. You can see it in the most primitive vertebrates, and even in non-vertebrates animals. There is a post here that gives some approximate numbers. For instance, 60% of your genes are directly related to the genes in a fruit fly.
I find this similarity in the DNA sequences between organisms truly amazing, no matter how many times I see it.
As Austin says, humans evolved very slowly. Ancestral Adam (the source of the Y chromosome in modern male humans) probably lived about 142, 000 years ago. Mitochondrial Eve (the source of the mitochondria or batteries that power our cells, and only inherited from your mother) lived about 200, 000 years ago. As we evolved, we developed a whole range of adaptations to help us survive the changing environment. As the land around began to change from dense woodland to grasslands, we evolved the ability to walk fully upright. Once we walked upright we were able to run more efficiently, which helped to hunt down prey, and to see more clearly over tall plant life. Walking upright also freed our arms to carry tools, food and our offspring. The development of tools could have contributed to the development of language, as we needed to communicate our culture from generation to generation. All of these modifications increased the size of our brains and made us into better problem solvers. This had the downside of making our heads larger and birth more difficult! This then led to changes in our pelvises and backs that we still have trouble with today.