The most common and most likely answer to this crossword clue is the 6 letter word REDDEN
I've seen this crossword clue in The Daily Telegraph
R
E
D
D
E
N
More clues leading to the result REDDEN
Buddy explains!
The straight part
We know that "Begin to look embarrassed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case begin to look embarrassed = redden.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 REDDEN mean?
REDDEN verb- To become red or redder.
- To make red or redder.