You can think about it iteratively. Supposed you had a weak base called "B-" which accepts a proton in this equilibrium.
HB <-> B- + H+
Now what happens if you add an acid (i.e. HA) to the solution from a strong acid? Then whatever B- is availabile will accept the H+ and form HB to oppose the change that was imposed and maintain chemical equilibrium (Le Chatelier's principle). Since its a strong acid, this will keep going until you run out of x- or H+.
Note that the reaction won't go to completion if you had a weak acid and weak base (you'll end up with an equilibrium). Think about why the above reasoning would fail in that case.