Cycles are repl.it's karma points. In simpler words - they're the total number of upvotes your comments/posts have on repl talk. You also get 5 cycles for best answer to questions in repl talk's ask board. (Mark this is the correct answer and refresh the page to see it in effect)
Cycles are the amount of total upvotes that a user has on their posts and comments. You also get five cycles at a time when your comment gets marked as the answer on Repl Talk Ask.
You can view this in action by clicking the check mark next to my comment and reloading the page.
Cycles are repl.it's version of karma on Reddit, upvotes on Stack Overflow, etc. Whenever someone upvotes a post, comment, etc. of yours, your cycles go up by 1. If someone accepts your comment as an Accepted Answer in the Ask community, your cycles go up by 5.
It's really just a way to show that you're active in the community :)