What is pairing energy of electrons?
Imagine a sky scraper where everyone wants to live on as low floor as possible, but landlord prohibits then to live more than 2 people per floor. These people will occupy lowest floors in pairs and the highest occupied level might have a single person or a pair. If someone moves out of the apartment on the 1st or 2nd floor, then someone from higher floors will move down to the vacant state. Electrons do the same. Floors are energy levels and the evil landlord is the Pauli exclusion principle.

Electrons are fermions. An electron's spin can be either +1/2 or -1/2. Two options. Two fermions (and, hence, electrons) cannot be the same. Electrons are the same if their spins are the same and they occupy the same orbital (Pauli Exclusion Principle). At lower temperatures (below red heat temperatures) electrons occupy the lowest energy level possible. 

As a result, each new electron occupies the lowest allowed level. Because their spins can take two values 2 electrons are occupying each level. If you populate each energy level by a single electron you will get a high energy excited state that well emit light and go to lower energy two electrons per level state. If you try to squeeze more than two electrons it will not work because of Pauli principle. So, since they want to stay in lower energy levels and have to obey the exclusion principle we have electron pairs. 



Bonus: here is why radicals recombine. For instance, why $\ce{2 H^. -&gt; H2:}$ 

If an electron is the last one it is occupying the energy level alone.
So, electrons would like to all occupy the lowest level, but only two per level is allowed. In radicals only one electron occupies a level. When radicals A$^.$ and b$^.$ meet each of them sit on half empty orbital attracted to a single nucleus.  It is energetically favorable to share the same orbital (form a pair, which they are OK with) and orient fragments so that they "fly" (have the highest probability) between two nucleus (A and B). Staying between two opposed charged ions is better than staying next to just one charged ion. And electrons are allowed to stay in pairs (Pauli excluded 3+ electrons staying together).