The most common and most likely answer to this crossword clue is the 2 plus 1 plus 8 letter word IN A NUTSHELL
I've seen this crossword clue in Financial Times
I
N
A
N
U
T
S
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More clues leading to the result IN A NUTSHELL
Buddy explains!

The straight part
We know that "to put it briefly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to put it briefly = in a nutshell.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.