Take a look at the pictures here (after Definition 1):
Paul's Online Math Notes. What you see there is that as you move from left to right on a
concave up graph, the slope is increasing (in the first example, from -3 to -2 to -1, and in the second, from 1 to 2 to 3, or whatever the numbers might actually be). Similarly, in the
concave down graphs, as you move to the right the slope decreases, from 0 to increasingly negative on the left, and from positive down toward zero on the right.
Possibly you are missing the fact that we are talking about
signed slopes, so that a line with negative slope becomes
less steep as the slope
increases (toward zero).
Does that help? If not, tell us more specifically what confuses you.