When two neutral species come together to form a coordinate bond, one of them provides both of the electrons in the bond.
In the case above, the nitrogen atom in #NH_3# has a lone pair of non-bonding electrons that can be donated to form a coordinate bond to the boron atom in #BF_3#.
#BF_3#, on the other hand, has only 6 electrons in its valence shell (two electrons from each of the 3 BF bonds). It can accept a pair of electrons in its valence shell to form a complete octet (8 electrons) in its valence shell.
Because both electrons in the new B-F bond originated with #NH_3#, the bond is called a coordinate bond. The resulting product, #H_3N-BF_3# is called an adduct.