I like to use the word "appropriate" for this:
appropriate (v): Take (something) for one's own use, typically without the owner's permission.
However this use feels more appropriate (pun intended) for things like artwork, musical compositions, inventions, etc. -- things with some material value.
The greedy Renaissance art school master was known to appropriate the works of his more talented students, so he could sell them at a higher price under his own name.
If you just tell people you were responsible for something when it was actually someone else, the verb is "to take credit for".
It was well known that the powerful politician would invariably take credit for his administration's successes, even when it was one of his staff who actually did the work.
Side note: There are many expressions for the opposite of this, to put the blame for your own mistakes on someone else. One of these is "to pass off".
However this politician (who shall remain nameless) always passed off failures as the fault of underlings, even when he himself was responsible.