What does it mean for a metric to have a curvature?

Question

I just considered reading this paper https://arxiv.org/pdf/1505.01994v1.pdf about spherical metrics. One of the first conditions to the metric is that it has "constant curvature 1".
Now I am asking whether someone could please explain to me how a metric can have a curvature since it is just an inner product on the manifold. Does it somehow induce a curvature in that context?