The green marks mean that you solved the puzzle "correct" the first time through. The red marks mean that you got it "incorrect" the first time (but of course you can still retry [and correctly solve it], but the mark will still be red until you quit, or reset the level past level 10 for a new session.
Source: https://www.chess.com/forum/view/community/solo-chess-1#comment-43776006
Hope that helps!
I’m kinda dumb, but I want some details on this red mark that appears after you’ve done that little game in solo chess.
How do I get all my games in the same color?