Explanation:
Imagine that f(x) was a road without any gaps. The road can be straight or it can be curved somehow.
Now imagine that a horizontal line is the border between two countries. The country up north is the positive country and the southern country is negative.
f(1) > 0 means we're in the northern country since f(x) is positive here
then f(3) < 0 means we're now in the southern negative country
Somewhere along the road we must have crossed the border at least one time. This must be the case because the road does not have any gaps in it and we cannot teleport. This is what it means to be a continuous function. To draw a continuous curve, you cannot lift your pencil up.
So because f(x) transitions from positive to negative, this means f(x) = 0 at some point at least once. So that's why there exists a c such that f(c) = 0
