"insist" as appearing in the sentence is an intransitive verb.
"insist" + preposition "on" is a kind of phrasal verb to be used to demand that something should happen.
~ insist on something (noun phrase)
~ insist on somebody (X) cannot be used, as there is no meaning at all!
~ insist on doing something (gerund phrase)
** We can put a noun or pronoun in front of a gerund to show who or what is doing the action in the gerund.
~ In formal writing, we must use possessive case;
~ In informal writing, it is acceptable to use a simple noun or an object pronoun instead; and
~ there is no difference in meaning in most of the cases between the possessive form and object form.
1. He insisted on my going with him. (formal writing)
2. He insisted on me going with him. (acceptable as informal use)
Both sentences have the same meaning!
However, it will not be the case for some transitive verbs as the meanings will be different.
For example, the transitive verb "like";
~ He did not like the woman standing in front of him at the parade. (participle phrase)
(= he didn't like the woman who was standing in front....)
~ He did not like the woman's standing in front of him at the parade. (gerund phrase with possessive case)
(= he didn't like the fact that someone (the "woman") was standing in front of him at the parade.)
Obviously, the above two sentences have different meanings.