Let's stay till nighttime and watch the fireworks.
Literal
Night [until-まで] be-and fireworks [obj-を] watch-[go-ていこう][invitation-よ].
いて is the te-form of いる ('be, exist'), chaining two verbs in sequence: 'stay until night, and then watch fireworks.' 見ていく uses ~ていく for 'watch and then go' or 'continue watching.' The final よ makes the invitation warmer and more inclusive.