How are neutral atoms converted into anions?

Anion is negative ion.
In an atom, there is equal number of protons and electrons, thus the positive charges balanced out the negative charges so an atom have no charge.

To form a ion, you must understand that only electrons in an atom are allowed to move, protons cannot move before it is held in the nucleus by strong attraction force.
Thus, to form a anion, the atom have to gain electrons. So that the ion have more electrons than protons and hence becomes negatively charged.

Extra:
Similarly, to form a cation (positive ion), an atom have to lose electrons so that the ion have more protons than electrons and hence positively charged.

An  atom  (ours, not carroll's) consists of a positively charged  nucleus, surrounded by one or more negatively charged particles called  electrons. the positive charges equal the negative charges, so the atom has no overall charge; it is electrically neutral. most of an atom's mass is in its nucleus; the mass of an electron is only 1/1836 the mass of the lightest nucleus, that of hydrogen. although the nucleus is heavy, it is quite small compared with the overall size of an atom. the radius of a typical atom is around 0.1 to 0.25 nanometres (nm), whereas the radius of a nucleus is about 10-6  nm.  *  if an atom were enlarged to the size of the earth, its nucleus would be only 200 feet in diameter and could easily rest inside a small football stadium.read more on -