Is there an in-universe reason for Anakin changing eye colour when turned to the Dark Side, or is there only an out-of-universe reason, i.e. it looks cool, and dark?
From the Wookiepedia article on Anakin Skywalker, his eyes changed color as a result of embracing the dark side of the force. This appears to be a common thing:
When Skywalker finally managed to climb up to the garden of the tower, he found Tano, brainwashed by the Son's influence, her eyes bearing the yellow hue of the dark side's taint.
During the self-inflicted onslaught, Palpatine went through a physical transformation as he struggled to intensify his dark side powers with his own life force, his face becoming hideously deformed and his eyes turning yellow.
This is confirmed from the entry for Dark Side of the Force:
Many darksiders experienced a change in the pigmentation of their eyes, as their normal eye color would transform into a luminous, sulfuric yellow. Often the skin would also lose pigmentation, becoming pale and mottled while veins became increasingly visible. The hair would also lose color, making it appear as though the darksider was aging rapidly. The user's voice would also be altered, becoming a few pitches lower and raspier.
At the end of ROTJ his eye color shifted back to their original blue.
However, his eyes were no longer the burning yellow of a Sith Lord, but the shining blue of his former self.
I think this is an extended-universe answer, but I hope it helps anyway. From what I've read the original Sith were aliens with red skin and yellow eyes. The Sith you mainly see in Star Wars are humans, and are either rogue Jedi or a descendant of rogue Jedi who turned to the dark side and joined the original Sith. The Sith race went extinct, but there are still rogue Jedi who called themselves Sith, have yellow eyes, speak the Sith language, etc. So when someone joins the dark side their eyes change to yellow eyes as a sign of them joining the Sith, not necessarily the dark side.