BF3 is a lewis acid and is thus an electron pair acceptor. The nitrogen on ammonia has an electron pair to donate (lewis base). The two react to form a new bond between the nitrogen and the boron.
The reason it is called a coordinate covalent bond is because both electrons are coming from one atom. In a normal covalent bond each atom donates one electron to form the bond. In this case BF3 has no electrons to donate and nitrogen has two. The boron can form a tetrahedral electronic geometry, but in doing so, shares the ammonia lone pair. The boron ends up with a formal 1- charge and the nitrogen with a formal +1 charge.