They cheated (sort of)
A great deal of that show consisted of computer animated dinosaurs (superbly done, to be sure) superimposed onto live filmed backgrounds. Many tricks were done like using a submerged device or something to make a splash in real water where the dinosaur would later put a foot. This is similar to what was done in the first Jurassic Park, where someone would use a rope to sway a tree, and later superimpose an animated dinosaur pulling on that branch.
Much of this can actually be done fully inside computers nowadays, but it would be much more difficult.