I'm note quite sure I understand the difference between an adverb and an adverbial participal. For example, in this sentence is there a real difference between "atende" and "atendante"?

Ili rigardas min atende, atendante, fakte, ke mi precizigu miajn dezirojn. (this comes from "Lasu min paroli plu" from Claude Piron).

Regular adverbs usually correspond to English words ending in -ly and usually modify a verb or adjective. But the adverbial participle needs to be attached to the subject of the sentence and describes some action done by the subject.

It's hard for me to explain this abstractly, so maybe some examples will help clarify the issue:

La biciklisto, vidante ke la vetero estis malbona, decidis ne veturi. = The bicyclist, seeing that the weather was bad, decided not to travel.

Note that you can't replace "vidante" with "vide" because the latter means "visually". So on the same lines, I'd translate your example sentence as:

"They looked at me expectantly [literally "waitingly"], indeed waiting for me to clarify my desires."

The second part of the sentence (after "ke" ) needs an adverbial participle to link it to "ili". Simply saying "atende" is not enough since "atende" only modifies the verb "rigardi".

(Hope this made things clearer. If not, I can try to reexplain.)

So adverbial participles are used at the beginning of a phrase right?

Sciante ke estos malfacila ekzameno morgaŭ, la studanto decidis ke si devus studi.

Knowing there will be a hard test tomorrow, the student decided that he should study.

Aaron94:So adverbial participles are used at the beginning of a phrase right?

If ke follows the participle, then it pretty much needs to be at the beginning (I can't think of an example where it wouldn't). Otherwise, it's permissible to put it at the end of the phrase, for example in:

La ĉiean detruadon vidinte, l' venkita reĝo ege malĝojis

But to my ear, this word order sounds almost poetic. In any case, I see it infrequently in regular usage.