So, water reacts with hydrochloric acid in the following formula
H2O + HCl —-> H3O+ + Cl-
We can visualize that when the two react, the hydrogen ions is taken on by the water molecule. This satisfies one of the definitions for a base
Bronsted acids = anything that donates a proton (H+ ion)
Bronsted bases = anything that accepts a proton (H+ ion)
So, as we can see, that is exactly what is happening. The Cl- and H+ detach and then the water takes on that extra H+.
H3O+ is what we call a hydronium ion