The most common and most likely answer to this crossword clue is the 9 plus 4 letter word QUOTATION MARK
I've seen this crossword clue in The Daily Telegraph
Q
U
O
T
A
T
I
O
N
M
A
R
K
Buddy explains!
The straight part
We know that "One starts a speech that another puts a stop to" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one starts a speech that another puts a stop to = quotation mark.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.