You are right to claim that erwerben calls for accusative case, and you are also right in stating that the accusative of seine Kenntnisse would be seine Kenntnisse. Seiner Kenntnisse is a genitive. But this genitive is not directed by the verb. The accusative phrase demanded by the verb is not seiner Kenntnisse, but actually viele seiner Kenntnisse.
Inside this accusative phrase (viele seiner Kenntnisse), seiner Kenntnisse is a genitivus partitivus, a genitive which indicates a relationship of being a part of something. viele seiner Kenntnisse translates to many of his skills. So, here we are talking of a part of his skills. In German (as in Latin) the genitive case is used to express this relationship.
Other examples of genitivus partitivus in German are:
- ein großer Teil der Gruppe ("a big part of the group")
- keiner meiner Freunde (”none of my friends", none is also a part)
- die Gesamtheit aller Menschen ("the totality of all humans", everything is also a part)
- zehntausend der Philister (tenthousands of the Philistines)