What keeps the sky from being lit up by stars at night ?

1  Human generated ambient light. Nearby lit areas are easier for our eyes to catch than more distant light. The nearby light drowns out more distant light which is why many call it light pollution.

2  Time. Light has a speed limit. While some of it is making its way towards us a lot of it is still on its way here (some of the stars that we see light coming from have actually burn out). Every star is being shown to us at a different time period.  We only see light from parts of the universe that are less than 15 billion years old.

3  Olbers Paradox states that the universe expansion is what causes light to be redshifted which lowers its frequency and energy.

Assumption:  A bright light should be seen in every direction.  Objects farther away are fainter but more numerous which should yield the same overall brightness.

Resolved !  The expansion of the universe known the cosmological constant, resolves the paradox. As the universe expands, the light waves are stretched out reducing the energy sent out from galaxy to another. Also, the time to receive the light is lengthened over the time it took to emit the photon. Because luminosity = the energy/time, the apparent brightness is lowered enough by the expansion to cause the sky to be mostly dark.
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4  Light from distant stars is less intense because the photons being emitted are emitted over a larger sphere area, fewer enter people's field of vision. There's also gravity (curves space) and interstellar gas like hydrogen and helium (blocks light).  General relativity shows how gravity is a curvature in spacetime.

5  Distance !  proven by the inverse square law

Why does a planet like Earth appear to not fall ?

Its orbit (which contributes to the magnetic field) gives the Earth angular momentum. The earth travels at enough speed while in orbit so that it doesn't fall into the sun, but not so much speed that it gets flung into deep space. Ironic since the earth's gravitational force is also being affected by the sun's mass. In the bigger picture (galaxies) there's also dark matter that exerts a gravitational force that gives larger clusters of bodies their integrity.