When you think about functional programming, you might think of banning mutation, or of functions without side effects. But in my opinion, the fundamental idea of FP is equational reasoning. That is, this code:
x = foo() y = bar(x) z = bar(x)
Should mean the same thing as:
y = bar(foo()) z = bar(foo())
Put explicitly: If you write x = foo(), that means you should be able to replace any occurrence of x with foo() without changing the meaning. That you can't have mutation or side-effects follows from this principle.
  1. If foo() had side effects, then the first example (where we call foo once) would have different behavior than the second (where we call it twice).
  2. If bar(x) could mutate x, then changing bar(x) to bar(foo()) would change what your program means.
By reducing what you can do, we allow a way of reasoning about programs that would otherwise be basically impossible. Math traditionally uses equational reasoning a lot, and not for nothing – it's really useful!

Can we still have effectful programs?

I'm going to be using a Rust-inspired syntax for this post, where you can write something like this to describe an enum with two variants:
enum Message { Quit, Move { x: i32, y: i32 }, }
So you could create a value of type Message by writing Move {x=3, y=5} or something like that.
Also, you can define anonymous functions with syntax like |x| x + 10. That describes a function that takes a value x and returns x + 10.
With the restrictions of functional programming, it's not obvious how to express programs that, y'know, actually do stuff. But it's not as hard as it seems. You just have a special variable, called main or something like that, and allow the user to set main to what I'll call an "action value" that describes the behavior of a program. What could this data structure look like?
  1. Start out with an action value that doesn't really do anything, like Done.
    If you write main = Done, your program will naturally immediately terminate without doing anything.
    So we're starting out with:
    enum ActionValue { Done }
    Ideally we would want to be able to write more interesting programs than that!
  2. Let's extend our ActionValue type a little bit:
    enum ActionValue { Done, Print {to_print: String, then: ActionValue} }
    Now you can write main = Print {to_print="Hello world", then=Done}, and your program will print Hello world and then be done!
    Or you could write:
    main = Print { to_print = "Hello world", then = Print { to_print = "Goodnight moon", then = Done } }
    And that would print Hello world, then print Goodnight moon, then be done. Ok, we're most of the way there, but we still need a way to write programs that take input from the outside world!
  3. Let's extend our type one final time.
    enum ActionValue { Done, Print {to_print: String, then: ActionValue}, Input {then: String -> ActionValue} }
    Now Input's then parameter takes a function of type String -> ActionValue, instead of a plain action value like Print's does. Which means we can write something like this:
    main = Input { then = |what_the_user_typed| Print { to_print = "Hello, " + what_the_user_typed, then = Done } }
    First, the computer sees Input, so it gets a line of input from the user. Then it passes what the user typed to the then parameter, and executes the action value it returns. So the result is the classic program that allows the user to type their name and greets them.
That's all we need! So why do languages like Haskell have their dreaded "monads"? Turns out all they do is simplify the construction of action values. (They're useful for other things too, but that's why they were introduced.) And it turns out with an sufficiently advanced compiler, it's not even that hard to convert an action value to pretty-efficient imperative code. So we gain equational reasoning without sacrificing any expressive power or even very much performance. Neat!

I don't know why people try to explain the IO monad without explaining this first

Monads often are just a convenient way to build up action values. In Haskell, the end result is that you can write code like this:
main = getline >>= (\what_the_user_typed -> print ("Hello, " ++ what_the_user_typed))
>>= is a built-in operator for combining action values, or anything else with a monad structure.
Or even
main = do what_the_user_typed <- getline print ("Hello, " ++ what_the_user_typed)
And you'll get something that's semantically identical to the above. The <- works like an =, except it signals that equational reasoning doesn't apply to this value. You can't replace what_the_user_typed with getline - your program won't compile. That's not a design feature that had to be added – it's inherent to the world of monads!