I would say first that monads are useful in that they offer a level of abstraction that can be used to describe lots of different algebraic phenomena. That is, categories of groups, abelian groups, rings, commutative rings,... can each be described equivalently as categories of algebras over some monad acting on sets (see Lawvere Theory for more examples). There are monads that don't arise in this way, but typically most examples are pretty close to this. So in one sense the theorems that are true in all of these situations are most naturally proven in the language of monads and their categories of algebras.

One of the other ways they show up in 'real math' is via the following: there is always a forgetful functor from the category of $T$-algebras in $\scr{C}$ to $\scr{C}$ with a left adjoint, the free $T$-algebra functor. This forgetful functor must also preserve and reflect certain coequalizer diagrams. In a precise sense (see Monadicity theorem) we can identify whether or not a functor is a forgetful functor from a category of algebras over some monad by these properties. This gives a useful criterion for seeing whether or not something is in the image of such a functor and what are the maps between such objects. This (or its dual formulation) gives descent type theorems, which is probably what algebraic geometers care most about. These theorems will generally tell you something like when is the category of sheaves of some type equivalent to a category of descent data of a very particular form.

I would recommend looking at Borceux's 'Handbook of Categorical Algebra' Vol. 2 Ch. 4 for more details. I'm sorry that I don't have a more geometric reference at hand.