Well, that is simple.
A carbon makes 4 bonds, a H or a halogen (X) makes 1 bond, N makes three bonds. So in a molecule with C Carbons, H Hydrogens, X Halogens and N Nitrogens, we will have (4C+3N+X+H) "bonding capacity".
Assuming there are no rings, no double/triple bonds, we must have (C+H+X+N-1) bonds in between these atoms [think like how many dashes you can draw between n o's: o-o-o -two dashes for three o's apparently].
Since each bond uses one "bonding capacity" from each of the two atoms its binding, (C+H+X+N-1) bonds are using 2*(C+H+X+N-1) bonding capacity. So the remaining capacity is:
(4C+3N+X+H)-2*(C+H+X+N-1) = 2C-H-X+N+2
If this is larger than zero, it means our molecule is unsaturated, i.e. still has a capacity to make bonds. And our assumption ["there are no rings, no double/triple bonds"] is wrong. This bonding capacity is used in forming new bonds either in rings or in pi-bonds. Since each bond uses two capacity -one from each side- we should have C-H/2-X/2+N/2+1 "unsaturated" bonds.