Not sure which "thingy" you refer to. The smaller lump has protection circuits. But the secret sauce to these emergency car starter power packs is in the big lump. The reason they can put out so much current (hundreds of Amps) despite being such small capacity (e.g. 8 Ah) is because they use lithium iron phosphate cells.
Lithium iron phosphate batteries have incredibly low internal resistance. This means they can dump their small capacity as huge amount of current (hundreds of Amps) in a short time (seconds instead of hours).
Let's compare a typical lithium-ion cell with a typical lithium iron phosphate cell:
- 2.6 Ah
- Impedance <= 80 milliOhms
- Max discharge current = 2C (which is 5.2 Amps)
- Nominal voltage = 3.7 Volts
A123 26650 (A bit fatter than an 18650, but similar capacity)
- 2.5 Ah
- Impedance = 6 milliOhms typical
- Max discharge current = 120 Amps for 10 seconds
- Nominal voltage = 3.3V
Looking at the max discharge current, there's a huge difference, and that is because of the very low internal resistance (or impedance).
Doing the math: if we draw 100 Amps for 10 seconds, how much capacity have we used? Assuming conversion is ideal, that is 100 Amp * 10 sec * (1hour/3600 seconds) = 0.28 Ah.
What? So that much current is only taking roughly 1/10 of the cell capacity because the time is so short! So we don't really need so much capacity to start a car: we need super low resistance for very high discharge current.
Another convenient thing with lithium iron phosphate is the nominal 3.3V and max charging voltage of 3.6V per cell. You put 4 in series and you get a charge voltage of 14.4 Volts. Great! That's right around where lead acid charging voltage is.
So put say 2 cells in parallel x 4 pairs in series and with 8 cells you can handle 200+ Amp discharges, 16 cells for 400+ Amps, at a voltage that is compatible with lead acid systems.
There's also electronics in there that protects from reverse polarity, etc, but what makes truly work are the very high discharge capability of lithium iron phosphate cells.