Why it works: you're using the same muscles (albeit only slightly different) when you lower yourself that you do to raise yourself so you'll be strengthening those muscles. Other exercises (bench press, etc.) can help with that, but ultimately if your goal is to do pushups, you need to be doing pushups as the motion/muscles that are used are going to be slightly different than in any other exercise. Once you're strong enough to do "proper" pushups, even just 3-4 GOOD ones, then you can drop the negative resistance reps and just do those, as that is going to be the BEST way to develop strength and ability. Your goal isn't to do a LOT if you want to get strong(er), it's to do them right (typically when you're training for strength you want to only be able to do 4-8 reps of whatever weight you're lifting, if you can do more than it's too light and you really won't be getting stronger as fast as you could be).

When your body is hungry it's sending a signal you shouldn't ignore; don't gorge yourself right before going to sleep, but going to sleep hungry is possibly just as bad.