If there is a relationship between Darwin's theories and modernism in literature or the arts overall, it is probably due to the fact that both movements sought to overturn previous ways of thinking about humans, history, and the universe. Darwin's theory of natural selection was seen as blasphemy by the religious people of his time (and later). It was an explicit rejection of belief in the scriptural account of the world's creation. This displacement of religion by another narrative reinforced the secularism that was already common among nineteenth-century intellectuals, including many in the arts.

It's obvious that the rejection of religious belief became even more widespread in the twentieth century, the age of modernism. It would be inaccurate to attribute this entirely to Darwinism, but Darwin's theories, as well as other scientific discoveries, tended to accelerate the secularist trends which had been occurring as far back as the eighteenth-century Enlightenment. And the aesthetic of modernist literature was similarly a rejection of the previous age's Romanticism and its often idealized view of man and the world.

Darwinism reinforced intellectuals' belief that humans are not special, spiritual beings but simply part of the animal kingdom. In modernist literature, there is often a focus on the more primitive forces that animate people, rather than on mystical or spiritual factors. T. S. Eliot's early poetry, for instance, emphasizes the trivial and even ugly side of human life. Prufrock (of his poem "The Love Song of J. Alfred Prufrock") is an insignificant little figure, one of the millions of purposeless individuals who have no special value beyond their flesh-and-blood existence.

Whether Eliot himself and other writers were directly influenced by Darwin's theories is a point I can't claim to have researched, though it would be interesting to do so. Darwinism and its secular, naturalistic orientation were "in the air," were part of the culture at the time, and thus relevant regardless of any specific connection between Darwin's thinking and that of writers and others in the arts.

Charles Darwin was a biologist during the late 1800s.  His most famous publication is the Origin of Species.  In it, he describes his theory on how animals and species can adapt and change over a period of time.  He explained how an animal may have certain adaptations that allow it to be more fit for a particular environment.  Better fitness means a greater likelihood of survival, which is why it is called survival of the fittest.  Those animals then pass on their traits through the process of natural selection to the next generation, and what is seen after many generations is an overall change in the characteristics of a population of animals.  

Modernism basically describes a series of reforming movements across all disciplines (art, music, literature, science, etc.) that began occurring during the late 1800s and into the early 1900s.  In general, modernism throws away the traditional thinking of the 1800s, which is why Darwin can be placed amid modernists.  Previously it was thought that species do not change.  They have been static since their creation.  The other major change with Darwin is that he said humans were no different than any other species that is out there, subject to the same laws of natural selection and fitness adaptations.  Many contemporaries of Darwin did not appreciate that because it linked humans with what they considered the "lower animals."