Lifting is just a fancy word for (in this specific case, anyway) taking a concrete value and putting it some kind of computational context.

I have a value 1 :: Int. There are a lot of contexts I can put that concrete value in! That is, I can make lots of values of type m Int that represent different views on the same value.

Let's start by looking at how functors (types that admit a lawful definition of map) lift values.

map :: Functor f => (a -> b) -> (f a -> f b) -- equivalently (and more usually) (a -> b) -> f a -> f b

Normally, this is explained by saying "given a function a -> b and a value of type f a for some functor f, apply the function on each a in f a and give me the result wrapped up in f". Another way of explaining it (which I've written the type for preferentially) is "given a function a -> b, give me a function f a -> f b. We're lifting the entire function into whatever our functorial context is.

Examples:

1 :: Int
map (+1) :: Functor f => f Int -> f Int
map show :: Functor f => f Int -> f String

Just 1 :: Maybe Int

map (+1) (Just 1) = Just 2 :: Maybe Int
map show (Just 1) = Just "1" :: Maybe String

[1] :: [Int]

map (+1) [1] = [2] :: [Int]
map show [1] = ["1"] :: [String]

Skipping directly up to monads, we really just need pure :: Monad m => a -> m a (a function which does nothing but put a value into a monadic [technically applicative] context) and bind :: Monad m => m a -> (a -> m b) -> m b. To keep it simple, we'll just reproduce map.

-- We compose our function with pure to get their types into a shape that bind will understand
incM :: Monad m => (Int -> Int) -> (Int -> m Int)
incM = pure . (+1)

showM :: Monad m => (Int -> String) -> (Int -> m String)
showM = pure . show

-- worth noting, Just 1 and [1] are both the same as pure 1 :: f Int with f fixed at Maybe and [], respectively

bind (Just 1) incM = Just 2
bind (Just 1) showM = Just "1"
bind [1] incM = [2]
bind [1] showM = ["1"]

bind (bind (Just 1) incM) showM = Just "2"
-- bind (Just 2) showM

bind (bind [1,2,3] incM) showM = ["2", "3", "4"]
-- bind [2,3,4] showM

So you can see that this is a way of getting a value "out of" some context (like pulling the value of a Maybe or all the values out of a list), doing some sort of transformation on it, then wrapping it back up in the initial context; it also lets you chain these transformations, finally wrapping everything back up when you're done. flatmap is called that because bind for the list monad is exactly "map a function over this list and then flatten the list".

bind ([[1,2,3],[4,5,6]]) id = [1,2,3,4,5,6]
bind ([[1,2,3],[4,5,6]]) (map (+1)) = [2,3,4,5,6,7]

I'll attempt a more thorough explanation, let me know if this makes any sense.

so I've got a type that represents some operations I want to provide:

data Op = Plus Int Int | Mul Int Int

I can turn that into a Functor by swapping the concrete values for a type variable:

data Op a = Plus a a | Mul a a

I'm doing this because I want to be able to compose these operations together - I should be able to freely sequence them however I like. so I can pass Op values in for a and nest them as deep as I like. I can also write an interpreter for Op values by breaking it down by cases and doing the obvious thing:

eval :: OP Int ->Int
eval (Plus a b) = a + b
eval (Mul a b) = a * b

I give that type the obvious, dumb Functor instance, nothing special (exercise left for the reader). then, I can pass Op to a function (liftFree) that turns it into a monad:

liftFree :: Functor f => f a -> Free f a

(I'm going to skip the actual definition of Free as it's just a type out of the standard library)

so I can use liftFree to turn the basic operations on Op (Plus and Mul) into monadic operations that are allowed to use do-notation:

plus :: a -> a -> Free Op a
plus a b = liftFree (Plus a b) 
mul :: a -> a -> Free Op a
mul a b = lift Free (Mul a b) 
calculation :: Free Op Int
calculation = do
    a <- plus 2 3
    b <- mul a 5
    plus a b

foldFree then allows me to pass it an interpreter function that evaluates my Op and turn it back into a regular value (like the obvious one I mentioned previously).

foldFree :: Functor f => (f r -> r) -> Free f r -> r
(foldFree eval calculation) :: Int

BUT because I can pass any interpreter I want, I've decoupled evaluation from the definition of the actions I'd like to take. so I could, instead of using an interpreter that calculated the final value, pass in one that pretty-printed it instead, or does a dry-run, etc..

prettyPrint :: Op String -> String
foldFree prettyPrint (fmap show calculation) 

so I can define actions that do a bunch of crazy IO stuff when called with a regular interpreter and run them instead with an interpreter that just sequences the operations and their arguments such that I can unit test that code without doing a bunch of mocking, etc.

I use a version of this trick wherever I can get away with it, even where I can't actually give a monad instance (like rust), because the decoupling alone is super powerful.