First, the output of a reaction is not necessarily less dangerous or at least as dangerous as itsit'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.