One of the consequences of the existence of magnetic monopoles is the Dirac quantisation condition: if we denote the charge of the monopole by $g$ and the electric charge quantum $e$ it can be shown that
$$eg = 2 \pi n$$
for integer $n$. This means that both electric and magnetic charge are quantised to a discrete set of values. Electric charge quantisation is observed, but not explained theoretically in the Standard Model (though other explanations for charge quantisation are available in extensions of the Standard Model).
Other consequences would depend on the precise nature of the monopole in question. Dirac monopoles are fundamental particles like electrons, and wouldn't really change our understanding of physics aside from explaining charge quantisation, and providing a neat duality between the electric and magnetic Maxwell equations. However, if the monopoles appeared as solitons of a Grand Unified Theory, then probing the properties of the monopole would give us information about physics beyond the Standard Model.