Because running at full power, a typical fuel rod will only last for 20 (25 if you're lucky) before it decays to the point of no longer producing useable heat and subsequently becoming a serious radiation hazard.
Ah, but you forget, in space no longer usable fuel rods can be tossed overboard and never seen again, you don't need to store them anywhere or worry about their radiation.
And that's why your ship will get boarded for a "health and safety inspection."
Well the nuclear power plant will only be at full power for 20 years out of the 5000 year voyage, for the rest of the voyage, it will only be used to provide light and heat for the habitat. There is no other power source available in interstellar space until the discovery of fusion.
A molten salt reactor is probably going to be better than a conventional nuclear reactor, for a host of reasons.
1) A conventional reactor runs at very high operating pressures, requiring a heavy containment vessel, yet the containment vessel has to open to allow replacement of the fuel rods. That's very heavy and complicated.
A molten salt reactor can operate at atmospheric pressure, so the reactor vessel itself doesn't need to contain high pressures.
2) A conventional reactor has all the short half-life highly radioactive breakdown products trapped in the fuel rods, which not only causes problems with fuel poisoning, but the issue of a massive release of radiation during a melt-down.
Molten salt reactors allow the seperation of breakdown products from the fuel during operation because they boil out of the liquid fuel, making it easy to vent them somewhere else for storage, if even venting them overboard as a gas.
3) Having the reaction products boil out as a gas also eliminates most of the problems with reactor poisoning, an issue primarily with xenon-135, which has a 9 hour half-life and a huge
neutron capture cross section, potentially sucking up so many neutrons that the reactor won't restart.
Reactor poisoning isn't much of an issue unless you have to rapidly cycle the ship's power levels for maneuvering to avoid space objects, but if you might have to do that, then designing to avoid poisoning is critical.
Naval powerplants use highly enriched uranium (instead of very low enriched uranium like commercial reactors) because they have to guarantee full power operation for combat maneuvers regardless of the output power levels over the previous few hours.
But going that route means you've massively increased the cost of the fuel, and created further handling problems because it's much, much more fissionable than commercial fuel.
4) Conventional solid fuel rods will need to be reprocessed, which is technically difficult.
If you toss them overboard, you're throwing away 97% of your potential fuel, which means you have to store over thirty times as much initial
fuel on board.
The fuel rods, obviously, can't be stored too close together or you could risk a chain reaction, especially if a moderator is introduced (such as from a broken water pipe). So each future fuel rod has a shielding or storage space overhead.
Unnecessarily upping the mass and space requirements for your fuel storage by perhaps several hundred times isn't a good design, and the ship will have to be able to reprocess fuel anyway, because if it can't, it will one day run out of fuel without the ability to manufacture any more no matter how much uranium the crew can dig up.
5) If the molten salt reactor uses thorium, which is extremely likely, then you not only can reprocess the fuel on the fly, as part of the normal reaction cycle, but you can store tons of thorium as a giant lump, because it won't support a nuclear chain reaction no matter how pure it is. Thorium is also 100% fertile, so it can all
be burned up. With the uranium cycle, you've generally got a lot of excess U-238 to deal with.
6) Molten salt reactors can run at much higher temperatures, making them more thermodynamically efficient. That means more available power to the drive system for a given thermal output.
On Earth, where the heat rejection is limited by outside ambient temperature, they could hit roughly 50% efficiency in a single fluid design. Water cooled uranium reactors run at about 35% efficiency.
In deep space, both reactors could have their efficiency impoved by further stages using lighter gases with lower boiling points, but the molten salt reactor would retain the advantage.
7) Molten salt reactors have a much higher specific power density, which is the energy output per mass. Conventional nuclear reactors work well in ships, which ran pretty well with coal-fired steam piston engines, but molten salt reactors were first designed to power an Air Force bomber
. In aerospace applications, that's an advantage that's hard to ignore.
8) Conventional reactors have to be shut down for long periods to replace the fuel rods. That means you have to have multiple reactors to guarantee that power will always be available to keep the ship from freezing.
Molten salt reactors can circulate the fuel in and out as part of normal operations, so they never actually need to shut down. They can also be shut down, the fuel drained, and then refilled and restarted up to full power in just a few hours, as opposed to weeks or months with a conventional reactor.
9) Since their reactor vessels doesn't have to hold high pressures, they are thin and lightweight, which means they are vastly easier to store, move, or fabricate, and multiple vessels could be carried for inflight swapping if neutron embrittlement becomes an issue.
The reactor shielding
(the room's walls) doesn't have to be structural, so it never has to be swapped.
So neutron embrittlement is at least easier to cope with in a molten salt reactor that has to run for centuries, because fabricating a very thick, high pressure vessel is always difficult, or requires carrying a whole lot of extra steel.
10) Thorium is more abundant than uranium, and doesn't require enrichment, so any human colony using thorium just needs to mine it, instead of trying to build an isotope seperation facility.
Canadian CANDU reactors don't require fuel enrichment, but they do require a source of deuterium, which they extract from seawater. Though not technically difficult, it does require processing about 6,000 times as much water as would otherwise be required.
And if the destination solar system is much older (or derived from older source materials) the U-235/U-238 ratio will be lower, possibly preventing even a CANDU from running without some level of fuel enrichment.
A final note is that from an engine standpoint, you've got the initial and final acceleration phases (which burn fuel), and your coast phase where you keep the ship from freezing (which burns fuel). Once you've got hard numbers for the power requirements of those phases, you'd optimize for the minimal total energy consumed during the flight. If the deep-space heating and lighting is a huge demand and delta V isn't that expensive, you'd accelerate more to shorten the trip (100 years of lighting takes 50 times less fuel than 5000 years of lighting).