The most common and most likely answer to this crossword clue is the 7 letter word ROYALLY
I've seen this crossword clue in The Guardian
R
O
Y
A
L
L
Y
Buddy explains!
The straight part
We know that "with great style" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with great style = royally.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 ROYALLY mean?
ROYALLY adverb- In a royal manner; in a manner having to do with royalty.
- Excessively; thoroughly.