Boric acid is an acid because its molecule:

Boric acid [${\text{B(OH)}}_3$ ] is a weak monobasic Lewis acid. It does not act as a protonic acid (by releasing ${\text{H}}^{+}$). Instead, it accepts a pair of electrons from a hydroxyl ion (${\text{OH}}^{-}$) of a water molecule:
$${\text{B(OH)}}_3+2{\text{H}}_2\text{O}\to [{\text{B(OH)}}_4{]}^{-}+{\text{H}}_3{\text{O}}^{+}$$
Since it accepts ${\text{OH}}^{-}$, it effectively causes the release of a proton from the water molecule.