How do they do it?

How do particle engines work?
And how are clamation games made?

Again, you should use google or the search function on this site. There's plenty of particle engine source + tutorials around.
That being said, you could try this tutorial: http://www.blitzcoder.com/cgi-bin/articles/show_article.pl?f=gamegiako07072002.html

check the b3d samples couple of examples there

Particle system generally work, by spitting out lots of quads/sprites/pixels, each individually controlled. Typically, each particle is a type object.

Type particle
   Field x,y,z
   Field speed
   Field life_time
   Field timer
   Field alpha
End Type

Each particle that is generated is basically

p.particle = new particle
p\x = 10
p\y = 10
p\z = 10
p\alpha = 1
p\speed = 5

Obviously each particle would have random values to a certain extent. Simple update (move and check life time of particle) and delete the particle if it gets too old. :o)

particles can be done any number of ways, most involve emitters that create sprites when updated. Some more advanced systems use vertex modification with quads to do the same thing but faster.

As for claymation, there is no specific difference between it and a normal 2d game (I'd be impressed if you could manage 3d claymation). You have pictures of objects at various stages of animation, as well as landscapes etc. For soem it might be easier to do such thigns hands-on (plus it looks cool), as opposed to using photoshop or gimp.

I guess. 3D claymation would be cool.
Maybe I'll just stick to buying a particle engine.

Noel Cower's "Lotus" particle engine is very good. And it's free.