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As an OO developer, maybe I have difficulty seeing its value. What added value do they give? Do they fit in an OO world?
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A closure is when a function remembers and has access to variables in its lexical / outer scope even when the function is executed outside of its lexical scope. Closures are created at function creation time. Variables are organized into units of scope, such as block scope or function scope.
A closure is a programming technique that allows variables outside of the scope of a function to be accessed. Usually, a closure is created when a function is defined in another function, allowing the inner function to access variables in the outer one.
The concept of closures was developed in the 1960s for the mechanical evaluation of expressions in the λ-calculus and was first fully implemented in 1970 as a language feature in the PAL programming language to support lexically scoped first-class functions.
A closure acts as a mechanism for diving into the pool, scooping up the coins (i.e. variables and constants) it needs, and sealing itself, such that whenever its parent function (the pool) drains, it and the coins inside survive.
You can see it as a generalization of a class.
Your class holds some state. It has some member variables that its methods can use.
A closure is simply a more convenient way to give a function access to local state.
Rather than having to create a class which knows about the local variable you want the function to use, you can simply define the function on the spot, and it can implicitly access every variable that is currently visible.
When you define a member method in a traditional OOP language, its closure is "all the members visible in this class".
Languages with "proper" closure support simply generalize this, so a function's closure is "all the variables visible here". If "here" is a class, then you have a traditional class method.
If "here" is inside another function, then you have what functional programmers think of as a closure. Your function can now access anything that was visible in the parent function.
So it's just a generalization, removing the silly restriction that "functions can only be defined inside classes", but keeping the idea that "functions can see whatever variables are visible at the point where they're declared".
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