You are correct that extra is a preposition here. Although it can be an adverb, it has a clear object here that would not make syntactical sense otherwise.
This particular argument follows Spinoza's earlier assertion that a cause must either "be contained" in a thing's nature or be given "outside of" that thing:
denique notandum hanc causam propter quam aliqua res existit, vel debere contineri in ipsa natura et definitione rei existentis (nimirum quod ad ipsius naturam pertinet existere) vel debere extra ipsam dari.
My translation:
Finally it should be noted that this cause due to which something exists, either must be contained in the very nature and definition of that existing thing (namely that it pertains to that thing's nature to exist) or must be given outside of the thing.
I also think it is worth translating the line immediately preceding your quote, which introduces the point of his whole "twenty men" example:
His positis sequitur quod si in natura certus aliquis numerus individuorum existat, debeat necessario dari causa cur illa individua et cur non plura nec pauciora existunt. Si exempli gratia in rerum natura 20 homines existant...non satis erit...causam naturæ humanæ in genere ostendere sed insuper necesse erit causam ostendere cur non plures nec pauciores quam 20 existant quandoquidem (per III notam) uniuscujusque debet necessario dari causa cur existat.
Translation:
Having granted these things, it follows that if in a nature some certain number of individuals exists, a cause must necessarily be given why those individuals, neither more nor less, exist. If for example, in the nature of things there exist twenty men, it will not be enough to show the cause of human nature in general (in genere = in its genus) but in addition it will be necessary to show the reason why neither more nor less than twenty exist, since the cause of the existence of each single one must necessarily be given.
Based on this line of reasoning, Spinoza's train of reasoning in your quoted passage becomes easier to follow:
But this cause (through notes 2 and 3) cannot be contained in human nature itself, since the true definition of man does not include the number twenty and thus the reason why these twenty men exist and consequently why each exists, must necessarily be given outside of each individual, and as a result it must be concluded absolutely that everything whose nature admits of many existing individuals must necessarily have an external cause for its existence.
Perfectly transparent, no? No wonder Jeeves loved Spioza :)
