First, the vacuum of space isn't inside something else. Remember that a vacuum is a volume with nothing inside it; no gas, liquid, or solid, nada (technically the vacuum of space isn't perfectly empty, but it's close enough). The reason space is a vacuum is that matter, like those gases, liquids, and solids, has mass; things with mass attract other things with mass (that's what gravity is). Simply put, matter clumps together, which is why stars, planets, and other things form.

Since that matter ends up clumped together, the space left between those clumps is more or less empty -- in other words, a vacuum.

Second, the negative pressure of the vacuum by itself is not what causes a collapse. Remember how matter clumps together because of its mass? The more mass something has, the stronger its gravity, and the more firmly things are clumped up. Earth's mass is great enough, and its gravity strong enough, that the gases in the atmosphere squeeze things inside it very strongly; at sea level, the weight of the air that is above a square inch of paper, going in a column all the way up to the top of the atmosphere, is almost 15 pounds. The higher up you go, the less air there is above you pressing down, which is why air pressure gets less the higher you go. In addition to that, those gases want to fill any empty space (because gases expand), so if there's an area that they can expand into, they will.

Combine those two things, and the air around you is actually pressing quite strongly on everything around it. So if you take a water bottle, and suck all the air out of it, there's nothing inside it -- in other words, a vacuum. But because a vacuum is literally nothing, then there's nothing inside that bottle pushing back against the air outside it -- which, as we just saw, is pushing really hard against that bottle. If the bottle had water inside, it would be strong enough to resist the air pressure at sea level -- even if it had air from the room it would be strong enough, because that air would be at the same pressure -- but the plastic along isn't strong enough. So it goes squish.

What causes the bottle to collapse isn't the negative pressure by itself; it's the difference between the positive air pressure outside crushing the bottle, and the lack of pressure -- negative pressure -- inside.

Imagine two people leaning against each other with their hands. Suddenly one of them lets go and steps out of the way real fast. With nothing stopping them, the other person falls forward, squashing anything that's under them. Removing the air from the bottle is like that person stepping away.