I again recommend reading through the Nuclear Weapon Archive site I linked to earlier as it’s a great way of explaining these sorts of things in such a way that even I can understand it. It didn’t come down on tablets from Sinai, but it seems to me to be able to answer a lot of the questions here (at least until Stranger on a Train makes an appearance.)
But yeah, from it, the third stage is fissioned by higher energy neutrons from the fusion reaction. From wiki, D-T fusion yields a neutron with 14.3 MeV. This is comfortably higher than the energy of a neutron release by fissioning either U-238, U-235, or Pu-239. Those neutrons—for reasons I don’t really understand—lack the energy to reliably fission U-238 (or Thorium, as mentioned by smithsb). In other words, they can’t fission the material that you’re making the tamper out of. If U-235 grew on trees, you could use it, but U-238 is roughly 300 times as prevalent. So you use it, instead.
Fusion neutrons can fission U-238 though. Accordingly, the fusion reaction, while providing a good deal of energy on its own, is also responsible for fissioning as much U-238 as you care to cover the device with. Now, how exactly do you couple the energy from the primary fission stage (or boosted fission) to the fusion secondary so it can start fusing? That’s still a secret, as far as I know.
My question in my first post was whether you could take the energy from this fissioning third stage, and use it to fuse a much larger supply of fusible material. As it is, we can only make the primary so large—call it about a megaton, though the link hints that more is possible—and therefore we can only make the secondary stage so large. Call it roughly 50-100 Mt. We can use that energy to fission a tertiary stage, but if we want to start moving asteroids or other things that you’d want a 1 Gt yield, wouldn’t it be nice if you could get another fusion stage?
Ironically, due to militaries getting away from very large weapon yields, there’s a thought that there’s no need for a fusion stage anymore. IOW, you can get to the 100-475 kT yields through a fission or boosted-fission weapon, so why have a thermonuclear stage at all? The link goes through some safety concerns with having a large, pure fission weapon. For one, large amounts of fissionable nuclear material have the potential of forming an unintentional critical mass. Again, as smithsb noted, this is undesirable.