The most common and most likely answer to this crossword clue is the 4 plus 2 letter word EATS IN
I've seen this crossword clue in Financial Times
E
A
T
S
I
N
More clues leading to the result EATS IN
Buddy explains!
The straight part
We know that "Gets pizza, stays home" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gets pizza, stays home = eats in.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.