Why is nuclear waste more dangerous than the original nuclear fuel?
Typical nuclear power reactions begin with a mixture of uranium-235 (fissionable, with a half-life of 700 Myr) and uranium-238 (more common, less fissionable, half-life 4 Gyr) and operate until some modest fraction, 1%-5%, of the fuel has been expended. There are two classes of nuclides produced in the fission reactions:
Fission products, which tend to have 30-60 protons in each nucleus. These include emitters like strontium-90 (about 30 years), iodine-131 (about a week), cesium-137 (also about 30 years). These are the main things you hear about in fallout when waste is somehow released into the atmosphere.
For instance, after the Chernobyl disaster, radioactive iodine-131 from the fallout was concentrated in people's thyroid glands using the same mechanisms as the usual concentration natural iodine, leading to acute and localized radiation doses in that organ. Strontium behaves chemically very much like calcium, and there was a period after Chernobyl when milk from dairies in Eastern Europe was discarded due to high strontium content. (Some Norwegian reindeer are still inedible.)
Activation products. The reactors operate by producing lots of free neutrons, which typically are captured on some nearby nucleus before they decay. For most elements, if the nucleus with $N$ neutrons is stable, the nucleus with $N+1$ neutrons is radioactive and will decay after some (possibly long) time. For instance, neutron capture on natural cobalt-59 in steel alloys produces cobalt-60 (half-life of about five years); Co-60 is also produced from multiple neutron captures on iron.
In particular, a series of neutron captures and beta decays, starting from uranium, can produce plutonium-239 (half-life 24 kyr) and plutonium-240 (6 kyr).
What sometimes causes confusion is the role played by the half-life in determining the decay rate. If I have $N$ radionuclides, and the average time before an individual nuclide decays is $T$, then the "activity" of my sample is $$ \text{activity, } A= \frac NT. $$
So suppose for the sake of argument that I took some number $N_\mathrm{U}$ of U-238 atoms and fissioned them into $2N_\mathrm{U}$ atoms of cobalt-60. I've changed by population size by a factor of two, but I've changed the decay rate by a factor of a billion.
The ratio of the half-lives $T_\text{U-238} / T_\text{Pu-240}$ is roughly a factor of a million. So if a typical fuel cycle turns 0.1% of the initial U-238 into Pu-240, the fuel leaves the reactor roughly a thousand times more radioactive than it went in --- and will remain so for thousands of years.
But it has to be more stable
That's where you're wrong. Most of the decay products are much more radioactive than the $\rm U^{235}$ that was used in the reactor. Uranium is not very dangerous at all. I have held a uranium rod in my hand. Admittedly it was a) coated in nickel and b) $\rm U^{238}$ which is less radioactive than $\rm U^{235}$.
The energy released in the reactor is not the radioactivity of the $\rm U^{235}$. Instead, the energy is generated by an artificial splitting of the $\rm U^{235}$ nucleus by impact from neutrons. The reaction products have a smaller combined mass than the $\rm U^{235}$ had, and the difference in mass is converted into energy.
These fission products tend to be very unstable, so they decay rapidly and release a lot of radiation in the process. They include isotopes like strontium $\rm Sr^{90}$ and cesium $\rm Cs^{137}$. Both have a half-life of about 30 years. On top of that, $\rm Sr$ is taken up by the body as a replacement for calcium, so all its radiation is released inside the body.
As most of the fission products decay fairly rapidly, their danger also diminishes quickly. However, the word "quick" is relative. For example, 30 years is long in human terms but very fast compared to the half-life of $\rm U^{235}$, 700,000,000 years. Hence, especially the initial containment is crucial but, since other reaction products have half-lives measured in millennia, long-term storage is also very important.
First, the output of a reaction is not necessarily less dangerous or at least as dangerous as it's input. Take dynamite for example(*): glycerin is a rather harmless material; nitric acid is a strong acid for sure, but still not as dangerous as the resulting nitroglycerin (active element of dynamite) that results from the reaction of those 2.
In a nuclear reactor, input fuel is a mixture of mostly uranium 238 ($\rm ^{238}U$ a very mild radioactive material), 2-3% uranium 235 ($\rm ^{235}U$ which is more radioactive than $\rm ^{238}U$, though radioactively very mild when compared with other radioactive materials, plenty of then will result from the fission reaction or split of this nucleus), and others.
To produce energy, a nuclear reactor splits $\rm ^{235}U$ nuclei into some lighter elements (this is the source of power, not its radioactivity). Almost all of the resulting elements are radioactive themselves, with their own radioactive properties. This is only part of the origin of the radioactive materials of a reactor’s waste.
The other part appears from a process known as activation. By this process, previously non-radioactive materials from the fuel rod will also become radioactive.
Combined, the waste result of a nuclear reactor is far more dangerous than the input fuel. As a matter of fact, when the fuel is inserted into the reactor, workers handle it directly, just using special gloves (not necessarily too thick or with a lot of protective material as lead). However, removing it from the reactor must be done remotely.
(*) This is just an analogy. nuclear reactions are a totally different process from chemical reactions. Still, the point is, products are not necessarily safer than inputs.