Informally, one may say that one is asking for c to locally 'stretch itself out' as much as it can, subject to the (informally considered) unit-speed constraint. The idea is that if c:[a,b]\to M is (piecewise) continuously differentiable and |c'(t)|_{c(t)}=1 for all t, then one automatically has d_g(c(s),c(t))\leq |s-t| by applying the triangle inequality to a Riemann sum approximation of the integral defining the length of c. So the unit-speed geodesic condition as given above is requiring c(s) and c(t) to be as far from one another as possible. The fact that we are only looking for curves to locally stretch themselves out is reflected by the first two examples given below; the global shape of (M,g) may force even the most innocuous geodesics to bend back and intersect themselves.
