> Am I wrong in thinking this guy isn't/wasn't a very influential person, outside of Twitter and the people that stay on there 24/7?

Yes, you are wrong, he was the leader of the most powerful campus conservative movement group in the country, was an extremely prominent figure in right-wing media, to the point where he is a central figure in pop culture images of the right, and a central target for being too soft of organizing figures for even farthe-right groups.

> What change was the person who shot him hoping to elicit?

Motives for assassinations (attempted or actual) of politicial figures are often incoherent. Political assassins aren’t always (or even often) strategic actors with a clear, rationally designed programs.