This short overview of the battle of badr explains the details you are seeking. The Battle of Badr
As for the verse you are looking for, is this the one:
So let those fight in the cause of Allah who sell the life of this world for the Hereafter.
And he who fights in the cause of Allah and is killed or achieves victory - We will bestow upon him a great reward.
And what is [the matter] with you that you fight not in the cause of Allah and [for] the oppressed among men, women, and children who say, "Our Lord, take us out of this city of oppressive people and appoint for us from Yourself a protector and appoint for us from Yourself a helper?"
[Quran Surat An-Nisa' - Verses 74-75]