What is the secret sauce of polysynthetic languages?
Sentences in polysynthetic languages can express a bunch in less words than other languages, sometimes a whole phrase in one word. But how this really works in practice I find difficult to grasp.
I get having extra ways of including information into verbs: aspect, mode, multi-person agreement, evidentiality etc. These can cut down on adverbs and maybe other expressions, but they don't seem to me to give that extra oomph that would make polysynthetic morphology stand out to me.
I don't imagine that the subject, object, and verb would all be included in one morphological word normally, especially if the roots for those three cannot stand on their own. I don't imagine polysynthetic sentences are mostly one word (or are they?). On the other hand, if the extra synthetic morphology is just to replace some adverbs, than it sounds no different from a very synthetic languages.
So, how do they actually work in practice? What are really the things or type of words that are incorporated as inflections instead of separately said and what IS usually kept as its own word/root?