How can the contact points of jumper cables handle the hundreds of amps required to start a car?
Hint: resistance is not just a matter of cross section, but also of length.

Would you expect the jumper cables to work if they were say, 10 meters long? What about 100 meters? You got my point.

Even if the contact surface is very small, the effective length of the contact is very short. One could argue that having flat, rounded surfaces would lead to a better contact anyway, but this is not true. The battery contacts are dirty and oxidized, the pointy alligator clips, together with the strong spring, help the stranded driver to perforate said oxide and guarantee a good enough contact.