First, it signals the end of Bill and Mary's attempt at conversation, startling Mary into the present.
If the lights symbolize truth or revelation, then their sudden brightness represents the irrefutable passage of time and the impossibility of ever recovering or re-doing the past. That the lights run "the whole length of Fifth Avenue" further emphasizes the completeness of this truth; there is no way to escape the passage of time.
It's worth noting that the lights turn on right after Bill says, "You ought to see my kids" and grins. It's a surprisingly unguarded moment, and it's the only expression of genuine warmth in the story. It's possible that his and Mary's children might represent those lights, being the brilliant chains that link the past with an ever-hopeful future.
