When we blew in Nintendo cartridges, we weren't cleaning dust off of them that was causing the problem; we were spitting in them just enough for worn down connectors to work a little longer. Not intentionally, obviously, but that was the end result, and why it worked.
I think it is more likely that the act of taking the cartridge out and reinserting scrapes away at grime that is stuck on the pins, creating better contact with the connector in the console. Blowing into them likely did nothing.
That’s not the reason, it’s because the constant removal & reinsertion of the cartridge across attempts to get it working causes the pins on the inside of the console to scrape against the contacts on the cartridge enough to remove some corrosion and form a proper connection. Saliva and the blowing had little or nothing to do with it.
The proper method is to use contact cleaner, rubbing alcohol or an eraser to remove the corrosion from the contacts on the cartridge. This is basically the same thing but instead of scraping off the corrosion with the console cartridge slot pins, you’re removing it evenly and cleanly.