The most common and most likely answer to this crossword clue is the 13 letter word HOUSE-HUSBAND
I've seen this crossword clue in The Times
H
O
U
S
E
-
H
U
S
B
A
N
D
Buddy explains!
The straight part
We know that "Man staying at home" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case man staying at home = house-husband.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.