Initially, at least, there wouldn't appear to be any particular connection between Charles Darwin's theory of evolution and existentialist philosophy but first impressions can be deceiving. As a matter of fact, there are important connections between the two. This is not to say that there is any sort of causal relationship because existentialism didn't inspire evolution and evolutionary theory didn't help produce existentialism. The relationship between the two is more a matter of spheres of concern and interest.
Evolutionary theory is, obviously enough, about life - and so is existentialism. Indeed, existentialists typically try to distinguish their work from that of other philosophers by emphasizing the fact that they are concerned first and foremost with how a person is to live in today's world. More than that, however, existentialism is about the struggle to live. This, you may already know, is also the central theme of Darwin's works.
Of course, for Darwin the "struggle to live" was a biological issue dealing with how members of different species compete for resources and strive to reproduce. Existentialists, however, have found the similarity between this biological matter and their own philosophical work to be very interesting. To a degree, it has provided a certain level of scientific backing to their insistence that the focus on how people live is of the utmost importance.
A further connection between Darwin's work and existentialism is the manner in which evolutionary theory contradicts certain traditional assumptions about the nature of life. In the past, people assumed that each species was created with a fixed nature. Each was assumed to behave in a fixed and immutable way because that was how God created them and how they had always been since the beginning of time.
Darwin rejected this, arguing in his evolutionary theory that species actually change over time. In the struggle to survive, only those species which are best adapted to their environments survive while the others die. Through the ages, this forces species to change both their physical and behavioral characeristics in order to become better adapted. Thus, there is no "fixed nature" aside, perhaps, from the principle of change and survival.
Obviously this is quite compatible with existentialist philosophy. Most existentialists have argued that we aren't born with a fixed human nature which forces us to act in certain ways and prevents us from acting in other ways. Instead, what we usually see as our "natures" is actually a product of our choices sometimes even choices we don't realize we are making.
Thus, Darwinian evolution provides some scientific credibility in more than one way to the existentialist position that humans make themselves and remake themselves during their struggle to survive in their day-to-day lives. Not all existentialists are necessarily "staunch Darwinians," however. Although it would be a rare existentialist who rejected the truth of evolutionary theory, there are those who don't regard it as having any real bearing on their philosophy.