The other answers have given good accounts of the initial reasons why we first thought monopoles might exist. However, modern physicists who believe monopoles exist tend to use a different argument.
In 1981 't Hooft and Polakov showed that certain classes of gauge field theory have vacuum solutions that carry a magnetic charge. In plain language, this means that there are solutions to the equations that describe the theory that behave like magnetic monopoles.
The Standard Model, which is the gauge theory that fits best to experimental data so far, is not one of these theories. However, it can be shown that if the Standard Model is part of any higher gauge theory (a Grand Unified Theory), then this theory must admit monopole solutions. Furthermore, we expect that if this is true, the conditions shortly after the big bang would have produced monopoles in copious amounts.
This raises the question as to where all these monopoles are. Some theorists think that this is evidence that no Grand Unified Theory exists. However, it is also widely believed that cosmic inflation may have diluted the abundance of monopoles so there is practically zero chance of observing them in the present day.
There's a good review for non-experts here: https://physicstoday.scitation.org/doi/10.1063/PT.3.3328
And a more technical one here: