Common ancestry suggests that humans have common ancestors with monkeys. This common ancestor seems to have lived about 30.5 million years ago. It isn't that one species or group turns into another but the variation within the previous group in the past lead to all the successive species.
Humans are members of the following groups in descending order of size.
cellular organisms; Eukaryota; Fungi/Metazoa group; Metazoa; Eumetazoa; Bilateria; Coelomata; Deuterostomia; Chordata; Craniata; Vertebrata; Gnathostomata; Teleostomi; Euteleostomi; Sarcopterygii; Tetrapoda; Amniota; Mammalia; Theria; Eutheria; Euarchontoglires; Primates; Haplorrhini; Simiiformes; Catarrhini; Hominoidea; Hominidae; Homininae; Homo
Japanese Macaques are members of the following groups in descending order of size.
cellular organisms; Eukaryota; Fungi/Metazoa group; Metazoa; Eumetazoa; Bilateria; Coelomata; Deuterostomia; Chordata; Craniata; Vertebrata; Gnathostomata; Teleostomi; Euteleostomi; Sarcopterygii; Tetrapoda; Amniota; Mammalia; Theria; Eutheria; Euarchontoglires; Primates; Haplorrhini; Simiiformes; Catarrhini; Cercopithecoidea; Cercopithecidae; Cercopithecinae; Macaca; Macaca fuscata
Common ancestry doesn't suggest that humans came from monkeys but rather humans came from Catarrhinis and monkeys such as Japanese Macaques also came from Catarrhinis some time in the past. And where variation with species are minor, those small variations in separate groups over the last 30.5 million years have lead to greatly varied species such as humans and gorillas on one side of the divide and macaques and baboons on the other. The theory of evolution suggests that the slight variation between species such as the gray and red squirrels could become as great as the variations between the different primates. It isn't that we are related to monkeys, but rather that if you go back far enough, we are related to all life on this planet.
The process by which organisms change is called evolution, and the mechanism is natural selection. Given any population of crickets for example if allowed to reproduce such that every offspring were able to grow to adulthood and have more offspring themselves the entire world would be several feet deep in crickets within a few generations. Therefore there must be forces which are restricting the ability of crickets to carry out their natural exponential growth. Predation, access to food, access to mates, etc limit the cricket population. Now, if in this necessary struggle for life a change were to occur in a cricket's DNA which was beneficial to it's possessor in this struggle for life, that cricket and those crickets with that gene would do better than those organisms without that gene. If it were not useful it would either quickly die or be less successful in the struggle for life.
What once was very little variation, over eons becomes very large variations. So what once was several millions of years ago only a slight change within a species become a small change between two species such as gray squirrels and red squirrels and as time goes on and more and more beneficial changes are added into the genomes as they better and better adapt to different and changing conditions of life, independently and among the great tapestry of lifeforms and selective pressures.
Taken to the intellectual limits this suggests that all organism on the planet are related and explains the biology, genetics, design, development, distribution, and every other aspect of life on this planet.
Somehow I think it needs a toss in with the theory of evolution by natural selection. A sentence or two with some oomph that if thought about would explain the theory.
update: added a couple paragraphs describing the main thrust of the theory.
Generally evolution isn't just a theory but an algorithm for beneficial change in direction of the selection. So if variations occur within a species and any of those variations are, for any reason, beneficial to that organism it will increase at a rate greater than that of the average population. So whatever seeks to keep a population down will keep only a certain subset of the population down. The population which can avoid being naturally selected against will flourish. If there is a general ability (like that in DNA) to change and engineer very specific structures and organs and organisms then this process will spiral up creating organism of ever increasing adaptation, complexity, and functionality. Further as every species is undergoing this process they all will play off each other in complex and wonderful ways like newts so poisonous that they could kill thousands of people with a drop of poison and snakes that regularly eat them. Like cheetahs running quickly and gazelles running about as quick. Like the antlers of a deer that weigh so much that they make the deer far easier prey and the females who mate with the males with the largest antlers. It's a very simple process but, even a cursory look will show that such a process would work, and when we look at nature there are abundant artifacts that are only explained by exactly that process.
It's all a work in process. Whatever explain the theory in an understandable way in a pretty short read is certainly the most fit variation of this.