The activity of a solid is 1 by definition. Ksp is called the solubility product because it is literally the product of the solubilities of the ions in moles per liter. The solubility product of a salt can therefore be calculated from its solubility, or vice versa.
I think you just need to memorize. EK general chem has a shorter list of the following in the first 3 chapters(I think):
Rules
Salts containing alkali metal cations or ammonium are soluble.
Salts containing nitrate, acetate chlorate or perchlorate anions are soluble.
Salts containing chloride, bromide or iodide are soluble. (Except for Cu+, Ag+, Hg2+2 and Pb+2)
Salts containing sulfate or sulfite are soluble. (Except Ca+2, Ba+2, Sr+2 and Pb+2)
Oxide and hydroxide salts are insoluble. (Except with group IA cations and ammonium.)
Sulfide and hydrogensulfide sals are insoluble. (Except with group IA cations and ammonium.)
Carbonate and chromate salts are insuluble. (Except with group IA and ammonium cations and ammonium.)
Edit: Actually I have no idea. To me this predicts that PbCl2 shouldn't be included.
So these are solubility rules. These are general solubility patterns used to tell if a salt is soluble in water. However, there is no such thing as something being completely insoluble in water. Every molecule has a certain degree of solubility in water, even if it is extremely small.
So, what is solubility? It is an equilibrium process between species in solution, and species not in solution (solids usually, although it could be a liquid or a gas).
Ok, so what does this equilibrium look like (in general):
Let's say we have a molecule X:
X (s) <----> X (aq)
The Ksp is just the equilibrium constant for this equilibrium, and since solids have activities of 1, the Ksp is just the concentration of X (aq). So that is the simplest case.
Most salts tend to ionize when they go into solution, which is why we commonly see a solid, which is one species, dissociate into two (or more) ions when they are in solution. However, in that case:
X (s) <----> Y (aq) + Z (aq) + W (aq) + ...
where Y, Z, and W are the different ions that X dissociates into. So our solubility equilibrium is the product of all of the aqueous species (and of course, raised to the power of their coefficient like usual).
So the only thing those solubility rules tell you is whether Ksp is high (high solubility) or low (low solubility).