How does the atmosphere help us survive, and what does the hydrosphere have to do with it?

"Help" is a pretty weak word for what the atmosphere does for us. If it were to suddenly disappear, we would all die within a few minutes.

We couldn't even hold our breath, as without atmospheric pressure you can't hold your breath, or your lungs will pop like balloons. Should you ever find yourself thrown into hard vacuum, you must exhale immediately. People have survived in hard vacuum for a few minutes, but no longer than that.

Thus, without the oxygen of the atmosphere, we would immediately suffocate. If that could somehow be prevented, we'd freeze to death in a few hours. If that could be prevented, we'd slowly cook from radiation. An atmosphere is absolutely essential for the survival of life as we know it; even organisms that don't breathe oxygen (such as plants and some bacteria) either breathe something else (like CO2) or depend on the atmosphere in other ways.

The hydrosphere we could last a bit longer without, maybe a few days before we all died of dehydration. Essentially all water on Earth is considered part of the hydrosphere, so without it we'd have nothing to drink.

Without the water vapor in it, the atmosphere wouldn't be as effective at absorbing radiation; this also means that the greenhouse effect would be weaker and the Earth would be much colder. We could probably still survive at that temperature, though (at least at lower latitudes), and the increased radiation penetration wouldn't be enough to kill us from ionizing radiation either.