Why is there the extra syllable "ent" in the noun?
According to the Digitales Wörterbuch der Deutschen Sprache it originates from mhd. ūfenthalt, "place for staying", which is an extension of mhd. enthalt, which means "resting/standstill/end" (in modern German it changed its (verb) meaning to "contain" or reflexive "abstain")
The prefix ent- in connection with verbs can express the beginning of a process (though this is now for modern German), so it could be related to a continued stopping, ie standstill. However, this is just my educated guess — etymology is often guesswork and usually not certain.
In the same dictionary there is no specific etymology given for aufhalten, so I guess this is just a straight forward modern combination of the prefix auf- with the stem halten. Auf- can express the result or completion of an action, so by stopping someone/something they are then halted. Again the caveat here is that this is for modern German.
So while Aufenthalt and aufhalten both ultimately derive from halten, they have taken different paths in their etymological life.