The most common and most likely answer to this crossword clue is the 9 letter word EXCITEDLY
I've seen this crossword clue in Financial Times
E
X
C
I
T
E
D
L
Y
More clues leading to the result EXCITEDLY
Buddy explains!

The straight part
We know that "with enthusiasm" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with enthusiasm = excitedly.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 EXCITEDLY mean?
EXCITEDLY adverb- In an excited manner.