Weaver is an anti-capitalist activist who despises the goverment and its stooges. He was incarcerated after a sit-in demostration outside The Exchange in Algonquin, Liberty City (maybe a parody of the Occupy Wall Street protest movement that started in New York City in 2011). Things went wrong at the demonstration and Weaver assaulted a cop with a fire extinguisher. Since then, he has made a conscious decision to "reject capitalist society" by living at Dignity Village in San Andreas.
When Trevor attempts to capture him at Dignity Village, Weaver will start shooting with a pistol and then attempt to run. However, he has no bodyguards and is relatively easy to stun. If captured alive, he tries to justify his surrender with an anti-government speech, then engages in a heated debate about anarchy and anti-establishments stances with Trevor. He eventually mentions Trevor's father, causing Trevor to fly into a rage.
This old-timer's one of them crazy anti-establishment types, lives outside the system in some kind of hobo camp. Might look harmless but watch yourself, he's got a couple of screws loose alright.