do I need to stack my influence to have a higher pressure on other nearby planets? 

If other nearby planets of enemy factions have a radius of influence against my planets, how can I push back?

The more influence you generate, the wider your empire influence circle. Near by systems can be converted via influence, if you have enough influence and i think special tech not sure. Only faction who can't do that are vodyani, they need are to exploit system.
To pressure enemy and win in score you have to do several things:
1. Bureaucratic embargo. Gives consistant pressure points each turn.
2. Have more score than enemy.
3. Attack enemy ships -> win -> refuse non attack pact. Gives pressure score once.
4. Invade enemy territory/system.
Basically doing aggresive moves and not giving a F.