The most common and most likely answer to this crossword clue is the 7 letter word PEEP-BO
I've seen this crossword clue in The Times
P
E
E
P
-
B
O
Buddy explains!
The straight part
We know that "that amuses baby" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that amuses baby = peep-bo.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.