The most common and most likely answer to this crossword clue is the 6 letter word YAPPED
I've seen this crossword clue in Independent
Y
A
P
P
E
D
Buddy explains!

The straight part
We know that "kept on talking" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case kept on talking = yapped.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.
What does YAPPED mean?
YAPPED verb- Of a small dog, to bark.
- To talk, especially excessively; to chatter.
- To rob or steal from (someone).
- Of a book: having a yapp.