How those servents get those class?

For the most part the way class is decided is based on the legend of the heroic spirit, so if their legend specifically deals with swordsmanship or they used a legendary sword in battle they would qualify for saber classification, this can be expanded the other 7 main classes. The 3 knight classes are mostly simple in that aspect, besides archer which is doesn't require a bow necessarily, but like you said high intuition. As for the cavalry classes they're a bit less inclusive, take semiramis who only poisoned one person qualifying as an assassin. And as for the others, Rulers typically have some piety but that's just kind of a coincidence, the real qualifier is if they have no worldy desire or wish for the grail which makes them neutral if summoned to moderate a grail war.

It's based on their legend/story, and it's pretty much about their NP.
Sabers used a sword or have a particularly famous sword as their NP.
Archers have a bow or projectile NP, they're pretty much the widest range among 'weapon' NPs.
Lancers have a spear. (or lance)
Riders have the widest range of all types. An animal or mount is typical, however commands of armies or ships/fleets also qualify you for Rider. It's more of a "hey this makes sense" than the other classes.
Casters were some type of mage or were associated with some form of magecraft. They could also be a scholar or be associated with something that magecraft covers. The authors are good examples of this.
Assassins are associated with, well, killing someone. Usually Hassans, who are associated with the name's origin. Admittedly, there's not a lot of these, so it's harder to say.
Madness Enhancement can be inflicted on purpose during summoning. But Berserker-only servants were insane or some other form of "crazy."
Rulers have no wish for the grail, or a purely beneficial wish, or something. Or Amakusa might be the only one who breaks the rule.
Avengers you know.
Moon Cancer is BB. It's only BB. It has to do with the plot of CCC.
Alter Egos are... alteregos. They're a personality split as a separate servant.
Foreigners are possessed by something stronger than a spirit. Abigail is possessed by Nyarly, Hokusai by something in the Abyss (and Hokusai's daughter by Hokusai).

Try the Type Moon wiki for better info.