I think you are trying to understand too much at once. Rather than trying to understand all of the "convex functions" page, focus on the first sentence or two and the first picture.

Convex functions: if you draw a straight line between the points (x, f(x)) and (y, f(y)), then f(z) is below that straight line if z is between x and y. (The green line is below the red line in the first picture.)

Optimization: find x so that f(x) is as small as possible.

Convex optimization: Same as optimization, but now you know f(x) is a convex function (e.g., the soft max).