When you do something vigilantly, you do it with a great deal of care and watchfulness. Your dog might vigilantly guard her special bone, growling a little when the cat walks too close to it.
Doing things vigilantly means acting with attentiveness. The adverb should make you think of a night watchman vigilantly watching a security camera or a Great Pyrenees vigilantly protecting a flock of sheep from predators. The adverb comes from its related adjective, vigilant, or "keeping careful watch." The Latin root of both words is vigilantia, or "wakefulness."