Anything that happens in an extreme or profound way is done intensely, like your practice of staring intensely at your cat in an attempt to read his mind.
A roaring fire burns intensely, and you'd be the first to admit that you love your dog intensely. This fierce and strong adverb works well for things that are literally or figuratively powerful, so you can study intensely for your geometry quiz or witness an intensely destructive earthquake. The Latin root, intensus, means "stretched, strained, or tight."