Call an integer bad if it cannot be represented as an integer combination of $a$ and $b$ with non-negative coefficients. There are $(a-1)(b-1)$ non-negative integers less than $(a-1)(b-1)$, and you know that all of the of the bad integers are among them. Take a look at a simple example. Suppose that $a=4$ and $b=7$, so that the bad integers must lie between $0$ and $17$ inclusive.
Notice that the numbers in each column add up to $17=(a-1)(b-1)-1$. A little experimentation will suggest that this is a general phenomenon: if $b=(a-1)(b-1)-1$ is the largest bad integer, and $0\le n\le b$, then exactly one of $n$ and $b-n$ is bad. If you can prove this, you’ll have as an immediate consequence that $\frac12(b+1)=\frac12(a-1)(b-1)$ integers are bad.
It’s easy to see that $n$ and $b-n$ cannot both be good: that would make $b$ good. Thus, you want to show that they also cannot both be bad. For this you’ll probably want to use what you know about the general solution to the Diophantine equation $c=ax+by$. If you get stuck, take a look at the proof of Lemma 3 in this paper by Mike Spivey; it isn’t exactly what you want, but it’s very close and should give you the ideas that you need to complete the argument.