Something that's done faintly is very weak, quiet, or almost imperceptible. When you're driving in the fog, you can often just faintly see the car in front of you.
A sick patient might call faintly for a nurse, and a distant sound, like the murmur of conversation from downstairs, can only be heard faintly. The day after you take down your Christmas tree, your living room might still smell faintly of pine, and a new acquaintance sometimes reminds you faintly of someone you know well. In the 14th century, faintly first meant "timidly," and then "feebly or wearily."