How do loose connections cause overheating, arcing and start a fires?
A loose connection causes a local increase in resistance. 

While the current will decrease, don't assume it will decrease much. For instance if you have 240v driving 12A through a 20ohm kettle, then one extra ohm in series will result in a new current that is still more than 11 amps.

The local heating will be given by \$I^2R\$, or 11*11*1 = 121 watts. Where the previous 2880 watts was being dissipated in the nicely water-cooled bulk of the kettle element, this shift of 121 watts into the loose contact is being dissipated in the tiny 1mm3 region of the loose contact surfaces, and will cause rapid and catastrophic heating.

An arc is basically a resistor, as far as power dissipation is concerned. It's the collisions between ions and free electrons, and their impacts into the contacts, that lose energy.