Lines nine and ten present reasons why other soldiers have gone to war, but lines eleven and twelve describe why the narrator has gone to war, "a lonely impulse of delight/Drove to this tumult in the clouds."
"Law" can be reasonably interpreted as the draft, which many countries had (or still have), a requirement that all able-bodied men (and sometimes women) must serve in the armed forces of their country.
"Duty" is what motivates many people to volunteer to serve in the armed forces of their country, a belief that everyone must do his or her part to help the country, really, a form of patriotism.
"Public men" is likely to refer to those politicians who exhort people to fight for their country, giving rousing speeches that do inspire some people to sign up for the military. I would imagine, for example, that Bush's speeches on the Iraq war or the war in Afghanistan persuaded many people to volunteer for the armed services.
"Cheering crowds" are for people who seek attention and praise, a bigger motivator than many people realize.