Space isn't actually holding the planets up at all. Earth is locked in a perpetual freefall, hurtling sideways at 67,000 mph just fast enough to constantly miss the Sun.
To understand why planets stay in place, you first have to discard the idea of "up" and "down." On Earth, if you drop a rock, it falls down. But in the wider universe, "down" does not exist as a universal direction. Down is simply the path toward the center of the nearest massive object. There is no cosmic floor for planets to fall onto.
Instead of falling down, planets are constantly falling around. In 1915, Albert Einstein revolutionized our understanding of gravity by proposing that space and time are woven together into a four-dimensional fabric called spacetime.
Imagine stretching a large trampoline tight and placing a heavy bowling ball in the center. The ball causes the fabric to sink, creating a deep funnel. This is what the Sun does to spacetime. If you take a marble—representing a planet like Earth—and roll it sideways along the rim of that funnel, it won't drop straight to the center. It will circle the bowling ball.
Earth's natural tendency is to travel in a straight line out into deep space, but the steep curve of the Sun's gravity well constantly bends that straight path into a circle. The precise balance of forward momentum and the inward pull of curved space results in a stable orbit.