The sentence is "passer à côté de [quelque chose]", which, in English, is "to pass by [something]". It means, in this context, "to miss out on something". It is an integral part of the expression; in your translation you translated that part by "close by" which is not correct, it goes with "miss out on".
The "[quelque chose]" is, here, the relative pronoun "lesquelles". In English you would translate it by "which" or "that". It is contracted with "des" to produce "desquelles = des + lesquelles", exactly like "de + les = des".
Simply using "que" would have been incorrect here: "passer à côté" can only take an indirect complement. You cannot say "je suis passé à côté toi", you must say "je suis passé à côté de toi" (I passed by you).
So a correct translation of the second part becomes:
But I have to admit that thanks to him I've been able to experience some things that I might otherwise have missed out on.
PS: Maybe the French sentence is easier to parse (but doesn't really sound good) if you split it in half:
Je dois quand même avouer qu'il m'a fait vivre des choses. J'aurai pu passer à côté de ces choses.
I have to admit that thanks to him I've been able to experience some things. I might have missed out on those things otherwise.