The most common and most likely answer to this crossword clue is the 6 letter word RIBBED
R
I
B
B
E
D
Buddy explains!

The straight part
We know that "Took a rise out of" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case took a rise out of = ribbed.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does RIBBED mean?
RIBBED verb- To shape, support, or provide something with a rib or ribs.
- To tease or make fun of someone in a good-natured way.
- To enclose, as if with ribs, and protect; to shut in.
- To leave strips of undisturbed ground between the furrows in ploughing (land).
- Having ribs.
- Intercalated with slate; said of a seam of coal.