It's kinda like a collet on a drill, think of the seat post as the drill bit:
as the top nut is tightened it pulls the segmented sleeve upward and tight around the post while forcing the outer sleeve apart and against the inside of the seat tube.
Here's what I've done:
take the clamp assembly (collet seems to be the accepted term) out of the frame and free of the seat post
disassemble, removing all oil and grit
thread all parts together making sure everything turns easily as it should
disassemble and oil everything that needs oil
oil preferred as grease can dry and gum it all up
adjust the assembly so it installs snugly into the seat tube and the seat post just barely fits in
install seat post
tighten top nut