I wonder what is the meaning of the bold expression:
They flew out to another state, on their own dime, to help brainstorm solutions for a colleague.
A dime is 10c (used specifically in the US or Canada).
In the context of your expression though, rather than being literal, you can think of this as just representing money in general and the use of "their" emphasizes that it's specifically them paying. So the expression essentially says "at their own expense", in contrast with the colleague (in this case) or their company paying for it, or reimbursing them.
You can use it in many expressions where someone pays for something personally, when someone else paying may have been possible or expected.