I think that John Rennie's is a good answer, specially due to the link given where it is shown the reasons why physicists expect the magnetic monopoles to exist. However I would like to stress that this is not simply the case of "explaining the non existence of something that we never saw". It is rather how to explain the lack of observation of something that it is expected to exist.
The strongest reason they are expected to exist is the fact they are predicted in Grand Unified Theories (GUT's), such as $su(5)\rightarrow su(3)\oplus su(2)\oplus u(1)$ or $so(10)\rightarrow so(6)\oplus so(4)\rightarrow su(3)\oplus su(2)\oplus u(1)$. They naturally appear as localized and stable solution to the equations of motions for the theory.
So a natural question to do is: Why do not we observe these theoretically predicted monopoles? Once they are predicted by a consistent theory we no longer can answer "they don't exist because they are not observed". We do need to explain why we do not see them even though the theory says they exist.
A simple but not interesting answer would be that the Universe did not show a spontaneous symmetry breaking patterns which allow for stable monopoles (such as the above ones).
A much more interesting scenario is the following one, called monopole problem. The monopoles predicted by GUT's are in general heavy and abundant. Its abundance is calculated to be far greater than the observed density of the Universe. This was one of the Alan Guth's motivation for proposing an Inflationary Universe. The idea is that these far too many monopoles were diluted during an infationary epoch.
A third explanation which have been proposed is that there was an intermediate phase during which monopoles anihilate themselves by means of flux tubes, which are other topological solutions predicted by GUT's.