The international community led by the US pressed Pakistan to deny
safe haven to terror groups operating from its soil and bring the
perpetrators of the Pulwama attack to justice.
This is saying that the international community "pressed" Pakistan to take certain actions, and the USA took the lead or was the most significant country in doing the "pressing".
In this context, "press" means "exert (political) pressure". "Press" would normally apply to things like coordinated diplomatic entreaties, whereas overt threats such as the risk of trade sanctions (or other retaliation for non-compliance) would generally require a stronger word than "press".
4a : to exert influence on : CONSTRAIN
4b : to try hard to persuade : BESEECH, ENTREAT
6a : to lay stress or emphasis on.
6b : to insist on or request urgently
[source: Merriam-Webster Dictionary]