You need the "of" in the sentence you provided because "of" is a preposition that describes the relationship. If you were to remove it from the sentence, i.e., "Let's hear some examples how to use the expression," you might get the point across (people will understand you) but the sentence is not grammatical. You need that connecting word (preposition) "of".
Whenever you are providing an example, you should probably say "example of". Only when you are talking about the word "example" generally would you not use "of" immediately following it.
One such sentence might be "That was a good example,"; "could you provide an example, please?";"I would like to seem some examples".
In all of those cases, you are not providing an example of anything, rather you are speaking about examples generally. However, say you don't know what a fruit is, and you want an example. I could respond "Pineapple is an example of a fruit."
I hope this helps!
***Also note in your title, you should probably rewrite to "When do I use the expression "example of' instead of 'example'?"
Prepositions in english are not easy and do require practice. Best of luck.