How can anions exist?

Lets say we have an nitrogen atom - it should have 7 protons and 7 electrons.

How can an nitrogen anion - lets say $\mathrm{N^{\,1\mathbf{-}}}$- even exist - shouldn't the 7 electrons in valence shell repell the extra one?

What force does hold the extra one electron in the valence shell? There isn't an extra proton in the $\mathrm{N^{\,1\mathbf{-}}}$ that would hold the extra electron.

Recall also that nitrogen has three (or five) valence electrons, rather than seven. The 1 s shell is full and is not considered part of its valency. The three 2 p electrons are the valence electrons although they hybridize with the 2 s electrons to produce the trigonal pyramidal structure of ammonia with its lone pair. The single anion $\ce{N^{-1}}$ could exist, but would not be stable because it puts four electrons in the p shell. The p shell would prefer to have three electrons as it does in the nitrogen atom or no electrons as it does in the $\ce{N^{3+}}$ cation.