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For a simpler explanation, because it sounds like that's what you want, I'll give you this:
If you have a function on a given interval, no matter how straight or how squiggly, that has some maximum and some minimum, it HAS to go through every single point in between the maximum and the minimum.
That means that If you have a function with a minimum of -2, and a maximum of 3, and the function satisifies the requirements of the theorem (is continuous, bounded on the closed interval), you know for certain that the function passes through every known point between the values -2 and 3.
What does this mean to you? If your teacher asks you on a test to prove that on the interval [-1,1], the function y=x has a value of y=0 at some point, you use the theorem to prove it. Looking at the graph, it's quite clear that it's a straight line, and that it passes through y=0, but in math you have to cite a theorem if you're going to make a claim like that.
Don't try to look too deep into it. It's as simple as it sounds. It's just a really basic theorem that describes common-sense behaviour of functions, with the purpose of introducing students to referring to known theorems to make claims/proofs.
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