You have to remember your strong acids and strong bases.
When you put a salt in water, dissociate it into its component ions, and then see how acid/base would react with it.
Let's look at NaCl
1. Dissociate it
Na+ and Cl-
2. React it
Na+ can combine with OH- to make NaOH... this is a strong base, so it will dissociate completely. That means it can't really "absorb" the OH- from solution, so Na+ will just chill in the solution and do nothing.
Cl- can combine with H+ to make HCl... this is a strong acid, so it too will dissociate completely.
Since the cation Na+ cannot absorb OH- and the anion Cl- cannot absorb H+, we would expect the pH of a NaCl solution do be about 7. This is what is actually observed.
Let's look at your CaCO3
1. Dissociate it
Ca2+ and CO3--
2. React it
Ca2+ can combine with two OH- to make Ca(OH)2. This is a strong base and will dissociate completely. This means it cannot "absorb" the OH- from the solution, so it will not modify the pH.
CO3-- can combine with two H+ to make H2CO3. This is a weak base, so not all of it will dissociate. That means in equilibrium we will have some H2CO3, some HCO3- and some CO3--. We've effectively "absorbed" some H+ out of solution, so there is less free H+ now. That means that the solution is more basic.
Since we could not remove OH- from solution (since it'd dissociate back into solution since Ca(OH)2 is a strong base), but we can remove H+ from solution by having CO3-- react with it... we're going to get a basic solution.
Let me know if this makes any sense.