There are two ways to explain why adding water cause more H2CO3 to dissociate
one is to again look at K (a1) - i.e. the K for the first dissociation step.
H2CO3 <=> H+ + HCO3 -
K = [H+][HCO3-]/[H2CO3]
as you can see if you add water top and bottom are diluted by the same amount. again let dilution factor be g (notice g < 1) then
Q = g^2 [H+][HCO3-] / g[H2CO3] = g * K
therefore ur Q, reaction quotient, is smaller, therefore eqm shifts to the RHS to make up for this decrease in K.
another way to think about is
H2CO3 is a weak acid, it doesn't dissociate 100%, but if you add more water, surely more will dissociate. Just like something wont dissolve 100%, if you add enough water, surely all will dissolve
this is a crude method though but easy to understand