What I thought was the key point was this:
Let's say you have a regular JS array. There is a reference to it in memory and it has a certain value. Now you change an element in that array. You've just mutated it. In JS, the reference in memory doesn't change, but now the value of what it's pointing to has. So in order to know if the value has changed, you need to do a comparison on every element on that array, which is expensive.
Now let's say you have a immutable array (such as from the Immutable FB library or Mori). If you change an element in that array, you get a new array and a new array reference in memory. In other words, if you check that the reference in memory is the same, you are guaranteed that the value is the same (in this case, the array). So checking for equality is fast and cheap because you are only checking the reference, not each value in the array.
One way this is important in React is that now if your component state (or entire app state) is represented by an immutable data structure, you can just check for reference equality in the shouldComponentUpdate() method when deciding to re-render that component. If the reference is equal, you are guaranteed that the data backing that component hasn't changed and you can return false, telling React not to re-render.
What's surprising is that people have made these data structures also very memory efficient. You can see David Nolan talking about how they work and memory comparisons here: https://www.youtube.com/watch?v=mS264h8KGwk
I personally like the safety benefits of immutability. You can give objects to functions and not worry about the functions changing the object. The only way to guarantee that otherwise is to clone the object, but that takes quite a bit of performance.
This is important for animations where you want to animate from the object's old value to the new value.