re: What actually causes inflation to slow down and stop?

Supply and Demand.

When Demand decreases, sales decrease, thus causing retail prices to drop to induce more sales. When Demand increases, supply can't keep up, and prices increase.

When people run out of money and quit buying stuff, the prices will stop rising and eventually drop.

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inflation happens when the money supply expands. Prices can’t go up forever if the money supply isn’t still going up, so then prices just stay where they are. Is this not true?

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Sort of. Inflation is a function of too much money in an economy chasing too few goods and services. We all tend to focus on the money supply in that aspect, meaning that the more money in an economy chasing a static number of goods and services, the more inflation we see.

As the old saying goes, it takes two to tango. In this scenario, the other tango-er is the amount of goods and services available. Make no mistake, money supply leads this dance but if money supply remains static while the supply of goods and services saw steady increases or decreases, that would be expressed through deflation/inflation (respectively) as well.

All that said, to get to your core question; if the money supply remains static then inflation can go up if the amount of goods and services decrease.