The most common and most likely answer to this crossword clue is the 6 plus 6 letter word ESCAPE CLAUSE
I've seen this crossword clue in The Guardian
E
S
C
A
P
E
C
L
A
U
S
E
Buddy explains!
The straight part
We know that "The key part of a Houdini contract" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the key part of a houdini contract = escape clause.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.