The phrase often implies subtlety; the speaker hints without stating directly. During the meeting, he was driving at someone's negligence without naming anyone.
Pro Tip 2/3
Context Clues
Pay attention to context, as it's key to understanding what the speaker is implying. Given the context of her speech, what she's driving at seems to be stricter regulations.
Pro Tip 3/3
Indirect Criticism
The phrase may be used to imply criticism or disapproval indirectly. Is he driving at my recent work being unsatisfactory?