When jumpstarting a car, you connect those big alligator clamps to the terminals of the batteries. The cables themselves can handle the current just fine, but the contact points between the alligator clamps and the battery is extremely small. I'd imagine a maximum of 4 teeth are touching each terminal. How do those contact points not simply explode under the hundreds of amps?
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.
Have you noticed how the clamps get hot after a jump-start?
That means power is lost, it ends up in those clamps. You're right the connection isn't ideal so power is lost but not so much that anything melts or explode. What is there to explode anyway? Nothing :-)
You wouldn't want to make that huge starting current flow for more than a minute as the clamps will heat up severely and the plastic insulation would melt.
But since we just want to start the other car, which usually takes much less than a minute, it is OK. The clamps can handle that huge current for a short time.
You seem to overestimate the effects of current on conductors. Let's take an AWG17 wire for example, which is a mere 1mm² of copper. Guess how much current it can take for 1 second before blowing up? Now check if you guessed right.
AWG 17 (1.04mm²) fusing current: 10s @ 99 A, 1s @ 316 A, 32ms @ 1.8 kA
Jump start clamps have a contact area of about 5mm², and the teeth are cooled by the thermal mass of the rest of the clamp, so handling a few hundreds of amps for a few seconds is not a problem.
AWG 10 (5.26mm²) fusing current: 10s @ 333 A, 1s @ 1.6 kA, 32ms @ 8.9 kA... AWG 11 (4.17mm²) fusing current: 10s @ 280 A, 1s @ 1.3 kA, 32ms @ 7.1 kA
Consider that copper is an excellent heat conductor. It is so excellent that, for example, you would have really hard time trying to solder the clamps to anything with even moderately powerful soldering iron.
Also, when you connect clamps there is typically a bit of arcing. The clamp actually gets welded a bit increasing the cross section of the actual contact with the battery leads.
Any condition of jaw connection, wire size, and other as mentioned above that causes a major reduction in voltage presented to the starting motor (series wound) causes greater current flow to produce the kW of power needed to turn the engine up to starting speed. Too much voltage reduction and the extra high current may destroy a perfectly good starter motor windings/set of brushes/commutator bars. Just a word of caution for any very low voltage starting situation.