Check out that Tackling the Awkward Squad that I mentioned elsewhere. It's about as lucid an explanation of the what and why as I've ever seen, by one of the original authors of Haskell.
With pure code you cannot really do much of practical value. You can perhaps compute values, solve relational or logic problems, do math, etc., but you only end up with results at the end of computations. Even that may require monads to print out or somehow follow a specific sequence, if the language is pure enough. Pure lazy code simply won't do anything, and evaluation order of lazy execution is variable, undefined even. Purity taken too far, won't do anything useful by itself.
So you use monads to add magic while starting off evaluations, often in specific sequences. They make up code blocks that are explicit about having side-effects. So are often used to separate pure code from impure code, as the magic monads can do could be anything one would like to add. So it's kind of like an impure addon to the existing language, but also by being explicit and more precise about the taints on purity. Each monad do different things, so are kind of like different addons.
However, monads may generally be used purely too. They are more general tools than just the famous IO monad. So the concept of monads are reused for abstraction. It's useful that so many things are solved by reusing the same general pattern. To explain all of that would require more context of the language itself, ie. Haskell. I'm sure somone with more experience has a short and easy explanation! :)
There are StackOverflow pages and such that have good explanations for those with a bit of familiarity with Haskell.
https://www.microsoft.com/en-us/research/publication/tacklin...