Why do atoms share electrons in covalent bonds?

Atoms share electrons in covalent bonds because in this way both atoms attain the more stable configuration of a noble gas (complete valence shell).

Explanation:

Covalents bonds are the chemical bonds that result when two atoms share the electrons of the valence shell (the outermost shell of the atoms).

The shared electrons may be count as belonging to the two atoms that form the bond, and so both atoms will gain the electrons that permit them to complete the 8 electrons in their outermost shell, which is the most stable configuration of the noble gases (2 in the special case of hydrogen and helium). This is the so called octet rule.

The covalent bonds are possible between elements that are close in the periodic table, meaning that their electronegativitiies are similar, and are typical of non-metal elements.

When two or more atoms are bonded by covalent bonds they form a molecule. Some examples of molecules formed by pure covalent bonds are: diatomic hydrogen, diatomic chlorine, and diatomic sulfur.

Since one atom of chlorine has 7 valence electrons it needs one additional electron to complete the 8 electrons (octet rule). Hence, two atomos of chlorine  may share one electron each, and so both atoms will count 8 electrons in the outermost shell, gaining stabilization both.