Why is earth warm and space cold?

The earth is warm because the atmosphere traps some of the sun's heat, whereas in space there is nothing to heat

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The temperature of space (except in the immediate vicinity of stars) is identical to the background electromagnetic blackbody radiation temperature left over from the big bang, which has cooled due to the expansion of the universe to about 3 degrees kelvin.

In the immediate vicinity of stars radiation, stellar winds, and stellar flares can heat space to temperatures between a few hundred degrees kelvin to a few million degrees kelvin.

As earth is very close to the sun, the temperature of the space enclosing earth is usually actually hotter than earth itself.