Use it when someone suddenly understands something important that they didn't see before. As he read the letter, it was borne in on him that he had won the scholarship.
Pro Tip 2/3
Emphasis on Impact
Shows that the realization has a strong, often personal, impact on the person. After the speech, the gravity of the situation was borne in on the audience.
Pro Tip 3/3
Not Instant
The phrase implies a process—understanding may take time to develop fully. While working on the problem, it was slowly borne in on her that there was no simple solution.