There are two possible literal meanings. 1. Montag's house burning to the ground. 2. Montag's entire city burning brightly as it is torched in the war.
There are more figurative meanings, or symbolic meanings, however. Burning bright could refer to Montag's entire life, way of living, perspective, and livelihood all burning brightly to the ground as he turns on it and escapes from the city. He chooses to rebel, and not in slight, subtle ways, he turns on Beatty and literally fires him, burning his rebellion for everyone to see. It is also symbolic of the phoenix that Granger mentions at the end; their entire city burned brightly to the ground, and will be born again, but this time hopefully built better, not to repeat the same mistakes. His relationship with Mildred officially burned in the last section. And, their society, so full and busy, that burned so brightly with activity, violence, and busyness, can also be tied to the symbol of "burning brightly".
"Burning Bright" also seems to be an allusion to William Blake's poem "The Tyger" from "Songs of Experience". The first stanza of the poem is:
"Tyger! Tyger! burning bright/In the forests of the night/ What immortal hand or eye/Could frame thy fearful symmetry?
The poem, about the tiger, the speaker presents the animal as some kind of strong energy that can be both a bring either a positive or negative energy. The poem's speaker cannot understand whether this energy is demonic or godlike. In this section of the novel, Montag is also questioning the forces at work in society, especially those responsible for the setting books on fire.