Why is Saber the "strongest class"? Characters in both Fate/Zero and Fate/Stay Night refer to the Saber class as "superior" or the "strongest" of all Servants but don't seem to explain why.

Is there something about how the summoning works that makes Sabers stronger or is it just that heroic spirits that qualify as Sabers tend to be more powerful ones?
The Saber class has the highest overall base parameters of any other servant class. It is part of the original three Knight classes. The other two being Archer and Lancer. The three Knight classes were intended to be used by the Three Families, Tohsaka, Einzbern, and Makiri. The four Calvary classes came later as they needed seven Servants. Basic parameter-wise, the three Knight classes are stronger.
Of course the strength varies depending on the individual Servant.
According to Rin in the UBW route - Day 4 Contract Established:

&quot;There is a change in one or two of the classes every Holy Grail War, but we have a basic lineup this time. The standard opinion is that Saber is the best Servant.
Each class has their own special characteristics, but the abilities of the Servants differ according to the rank of the heroic spirit summoned.&quot;

The prologue of Fate/Zero mentions that the Saber class made it to the final battle in the first three Fuyuki Grail Wars (and also the Fourth).
The Saber class skill Magic Resistance gives it an advantage against the other classes, especially against Caster, for obvious reasons. The weakness of the class is probably it's straight forwardness in attacking things head on, compared to other classes that can resort to other tactics with their skills. Like Assassin with it's Presence Concealment.

Base servant container stat comparison breakdowns for each class. Stats for the actual heroic spirits may very drastically, especially for the Berserker class.