The definition of computation monads (in sense of Moggy) is simple, but not trivial. It's deep since it generalizes many computation "phenomena": side effects, exceptions, etc.
Conceptually, a "pure" function is something as `A -> B`, so all information about its internal (e.g. side effects) lost: we put something of type A but we can only observe a result of type B.
The idea of computation monads is to attach the result type of the function with a "constructor" T, then the type T B is used to "restore" (or to reason about) the lost information.
In composing such functions, T X (for some result type X) is all information we need, then this explains naturally the third rule of the Kleisli triple `f: A -> T B; g: B -> T C`: A -> T C
Conceptually, a "pure" function is something as `A -> B`, so all information about its internal (e.g. side effects) lost: we put something of type A but we can only observe a result of type B.
The idea of computation monads is to attach the result type of the function with a "constructor" T, then the type T B is used to "restore" (or to reason about) the lost information.
In composing such functions, T X (for some result type X) is all information we need, then this explains naturally the third rule of the Kleisli triple `f: A -> T B; g: B -> T C`: A -> T C