Stars can emit their own light, but why can’t planets?

The simple answer is because planets are not hot enough, and that’s because they lack the huge internal energy sources that stars have: thermonuclear fusion. (They do emit a tiny bit of light, mainly infrared light, but nothing compared to the Sun.)
Essentially, the Sun shines because its surface is at 5800 K (around 5500 °C or 10,000 °F) and is thus “white-hot”. Anything that is hot enough will shine through the emission of black-body (thermal) radiation.
The surface of the Earth was perhaps at 1000–2000 degrees after it formed 4 billion years ago. When the surface was still molten it must have emitted a “red-hot” glow, just like molten metal just out of a furnace (like in the artistic impression below). But it has since cooled, and nowadays only its interior remains at temperatures high enough for iron and silicon to remain in a molten state.
The big difference between the two is that, once formed, the Earth gradually cooled off, slowly radiating the finite heat of its creation away into space, unable to generate energy on its own to sustain its surface temperature. Well, there’s the release of energy due to the decay of radioactive elements, but it’s nowhere as powerful an energy source as that of the Sun, so the planet still cooled quickly.
(In fact, the first physical hypothesis to explain how stars shine, proposed in the 19th century, was that they did so in the same way as the Earth or molten metals do, just by radiating away their stored thermal energy — the so-called Kelvin-Helmholtz mechanism. But calculations quickly showed that if this were the explanation the Sun could only shine at its current level for a few million years, not the billions of years we know it has from geological records.)
The Sun (and all stars) sustains thermonuclear reactions at its core which continuously release tremendous amounts of energy (by fusing hydrogen nuclei into helium nuclei). This lets it keep its very high surface temperature, and shine brightly, for billions of years.
Fusing hydrogen requires very high pressures and temperatures that exist nowhere on Earth (naturally). The enormous masses of stars (the Sun is a 300,000 times as massive as the Earth) mean they have very strong gravity, and that is enough to compress the core enough to reach the conditions necessary for thermonuclear fusion.
The Sun won’t shine forever, though. Eventually it will run out of hydrogen to fuse in its core, and will switch to fusing helium, but that will also run out pretty quickly. After that it will be left without elements to fuse, and will become a slowly cooling white dwarf (extremely hot at the beginning, upwards of 100,000 degrees).
Sirius B (the tiny dot to the left and below center in the Hubble photograph below), the companion of Sirius which is the brightest star in the sky, is an example of such a “dead star”. While it is currently very hot (10,000 degrees) it has no internal energy source and thus is destined to cool down and gradually emit less light until it no longer shines enough to be seen, as the Earth did a long time ago (Sirius B has much more stored thermal energy, though, so it will still shine for billions of years).

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