We can answer this from the perspective of the common ion effect, and the Ka. Think about dissolving salt in pure water - NaCl dissociates into Na+ and Cl-, and has a very high molar solubility, so you could dissolve a significant amount. At some point, however, you want be able to dissolve anymore NaCl because the solution is saturated - it can't handle any more Na+ or Cl-. However, what if you now try to dissolve glucose into the water? That's fair game, because the molar solubility reaction for glucose (s) --> glucose (aq) doesn't have any salt in it. For acids/bases that dissociate into ions, it's actually ok to think of them as salts. For the purposes of this post, let's pretend that solubility means dissociating into ions.
Ok, let's stick with our example of saturated NaCl solution for a bit. What if you wanted to dissolve solid HCl or NaOH in it? Those are both strong acids, and we expect them to dissociate completely, don't we? Well, it's not so simple. We already have so much chloride and sodium dissolved. Well, it turns out that HCl is more soluble than NaCl. So HCl will dissolve, but now you're adding Cl-, a product of both reactions, which will drive the NaCl dissolution reaction to the LEFT. We will precipitate NaCl.
In pure neutral water, we have H+ and OH- each in concentrations of 1*10^-7 due to autoionization. What if we had a pH of 2? Well, this would mean that we now have 100 mM H+ ions, and if our pOH is 12 then we have 1*10^-12 OH- ions. So having a high concentration of H+ in solution means a greater common ion effect for any (arrhenius) acids we're trying to dissolve. However, having such a low [OH-] means that our common ion effect for arrhenius bases is super weak, even weaker than pure water.
Essentially, just think about the reactions. Let's make an ice box.
HA --> H+ + A-
I: 1M | .1M 0
C: -x | +x +x
E: 1M-x | .1M+x +x
Ka is equal to the products over the reactants, and it's constant, and the only thing that's changing with the pH in this reaction is the initial value of H+. We know that 'x' is equal to molar solubility. Clearly, if that initial value of 0.1M were smaller (higher pH), then the x value would have room to be bigger, and we would get more dissociation of our acid.
To be clear, an acid can dissolve without dissociating. Many weak acids are highly soluble. Acetic acid (s) will readily dissolve in water into acetic acid (aq), but because it's a weak acid it won't undergo much dissociation, where it would go from acetic acid (aq) --> acetate + H+. If we increase the acidity of our solvent, acetic acid will dissociate less due to the common ion effect. If it dissociates less, that means we'll have MORE acetic acid (aq) in our solution (instead of acetate). If we have more acetic acid in our solution, then it makes it difficult to dissolve more acetic acid (s).
Conversely, we could have a very strong acid/base that dissolves very poorly. One example is Ca(OH)2. If you pour some solid calcium hydroxide into water, you won't really dissolve much, and will be able to see the white powder floating around. However, you can bet that any of the solid that DOES dissolve into solution will very readily dissociate. So Ca(OH)2 (s) has a low KSp (poorly soluble), but Ca(OH)2 (aq) has a very high Kb (very basic).
