It's not even about the effects, really. It's simpler to think of Monad as an interface for manipulating encapsulated data in a way which keeps it encapsulated.
The simplest way to encapsulate data is to wrap it using a constructor which we keep private to our module:
module M1 (Encap(), val1, val2)
data Encap a = mkEncap a
val1 :: Encap Int
val1 = mkEncap 5
val2 :: Encap Int
val2 = mkEncap 10
Other modules importing M1 get access to the Encap type, val1 and val2 but
not the mkEncap constructor. They can use val1 and val2 as-is, but they can't construct new Encap values or destruct existing Encap values to get at their contents. The problem is, there's not much we can do with this interface.
One way we can make this more useful is being able to apply some function to an encapsulated value. That's what Functor is for, so we can add this to M1:
instance Functor Encap where
fmap f (mkEncap x) = mkEncap (f x)
Now users of M1 can transform encapsulated values without being able to break the encapsulation. For example:
val1Plus7 :: Encap Int
val1Plus7 = fmap (+7) val1
val2Str :: Encap String
val2Str = fmap show val2
We're still pretty limited though, since there's no way to combine encapsulated values into new encapsulated values, or to encapsulate our own values. That's what Applicative provides, by letting us construct encapsulated values
without gaining the ability to destruct them, and by allowing encapsulated functions to be applied to encapsulated values (since functions are closures, this lets us gather up encapsulated values and combine them arbitrarily):
instance Applicative Encap where
pure x = mkEncap x
(mkEncap f) <*> (mkEncap x) = mkEncap (f x)
Now users of M1 can encapsulate and combine values, like this:
-- val1 + val2
val1PlusVal2 :: Encap Int
val1PlusVal2 = fmap (+) val1 <*> val2
-- New encapsulated string
val3 :: Encap String
val3 = pure "Hello world"
-- val3 repeated val1 times
val3Repeated :: Encap String
val3Repeated = fmap rep val1 <*> val3
where rep n _ | n <= 0 = ""
rep n s = s ++ rep (n-1) s
This is quite a powerful interface, but one thing we can't do is `collapse` double-encapsulated values into single-encapsulated values. That's what Monad provides:
join :: Encap (Encap a) -> Encap a
join (mkEncap (mkEncap x)) = mkEncap x
An alternative, but equivalent, definition is to allow calls to encapsulation-producing functions without encapsulating their result. Haskell's Monad is defined this way:
instance Monad Encap where
(>>=) :: Encap a -> (a -> Encap b) -> Encap b
(mkEncap x) >>= f = f x
These kind of encapsulated values turns out to hold effects without breaking the language, and these interfaces turn out to be powerful enough for general computation.