Learn Dutch: De timer gaat af en daarmee begint het spel. - The timer goes off and with that the game begins.

Daarmee (literally "there-with" or "with that") is the demonstrative form. Here it refers back to the entire preceding event (the timer going off). While ermee ("with it") is grammatically possible, daarmee is much stronger when pointing back to a whole clause as a trigger.

How does this sentence relate to the lesson's focus on substituting daar for er?

Imagine if the subject was indefinite: "with that, a game begins". That requires the existential er, making it begint er een spel. If you tried to use ermee, you would get en ermee begint er een spel. Dutch avoids this "double er" clash. The fix is substituting the first er with daar, giving you en daarmee begint er een spel. This sentence uses daarmee naturally, which is a great habit to build to avoid those traps.

Why is the word order begint het spel instead of het spel begint?

The second clause after en starts with the pronominal adverb daarmee. Because a word other than the subject occupies the first position in this main clause, the verb and subject must swap places, which is called inversion.

I know afgaan means "to go off" for an alarm here, but are there other meanings a B2 learner should know?