NH3+BF3→ H3N:BF3
From the reaction it is clear that BF3 is accepting the electron pairs from :NH3.
Although BF3 does not have a proton but acts as Lewis’s acid as it is an electron deficient compound.
It reacts with NH3 by accepting the lone pair of electrons from NH3, forming a co-ordinate bond and completes its octet.
Acids are electron pair acceptor and bases are electron pair donors. So, BF3 is a Lewis acid and NH3 is a Lewis base.
So, Lewis theory of acids and bases explains it.
Hybridization
Boron in BF3 is sp2 hybridized- as 3 orbitals i.e. 2s,2px,2py gets involved in bond formation.
whereas N in NH3 is sp3 hybridized with 4 orbitals - 2s, 2px, 2py, 2pz involved .