The slope of a graph at any point is defined as the derivative of the y-variable with respect to the x-variable. For a displacement-time ($s-t$) graph, the slope is $\frac{ds}{dt}$. By definition, the instantaneous velocity $v$ is the rate of change of displacement: $v=\frac{ds}{dt}$. Therefore, the slope of the $s-t$ graph represents velocity.