Theres a lot to the chemistry of tarnishing, but in short the main reason silver tarnishes is humidity and sulfur in the air. Its tough to give guarantees on how things will tarnish, but the typical course is if you have a low humidity environment with sulfur and no oils and salts from your grubby hands, you're more likely to produce silver oxide compounds that'll give you that pretty rainbow tarnish you see people selling for too much money. I don't know in particular what the specific chemical reaction is, but if you touch coins when they tarnish they're far more likely to end up black and dirty looking.

The aluminum foil method you use is sort of the reverse of electroplating. Instead of a plating solution you're creating an electrolyte solution with the baking soda, and theres a difference in electronegativity between silver and aluminum that creates a tiny voltage that makes the sulfur compounds on the surface of your silver break off and form bonds with the aluminum instead of the silver. I don't actually know if thats a good way to clean your coins or not. Its not abrasive, but the reason people don't use chemical dips for example is it will eventually remove the mint finish that comes with the striking process at the mint by their polished steel dies. I've dipped silver coins before melting them in harsh jewelry cleaner and the effects weren't visible. Without spending a lot of time researching exactly what constitutes damage to the finish of a coin removing the sulfur through electrochemistry may or may not do similar damage and it may not be immediate.