I think it's a typo. \int_a^x f'(t)dt should be f(x) - f(a), and that would be the fundamental theorem of calculus (or some corollary of it). It looks like either f(a+x) should be f(x) or the upper limit x should be x+a. The rest of the post deals with Taylor series, which are often written with f(a+x) for the function being expanded, so that might have mixed the author up when writing that. They don't directly use that expression elsewhere as far as I can tell, so it doesn't affect the rest of the post.

If it's f(x) = f(a) + \int_a^x f'(t)dt, then you can think of f(x) - f(a) as the area under the curve of f'(t) on the interval from a to x. For example, if f(x) = x, f'(x) = 1. The area of a function with height of 1 from a to x would be x - a, which is the same as f(x) - f(a) = x - a.