Learn German: Man behält die Vokabeln, indem man das Gehirn trainiert. - You retain the vocabulary by training the brain.

indem is a subordinating conjunction that expresses how something is done, translating to 'by doing' in English. Like all subordinating conjunctions (such as weil or dass), it sends the conjugated verb trainiert to the very end of its clause.

English says 'by training', but the German sentence does not use an '-ing' word. Why is that?

German does not have an exact equivalent to the English '-ing' form (the gerund) for this grammar structure. Instead of saying 'by training', German builds a whole new clause meaning 'by the fact that one trains' (indem man ... trainiert).

Why use behält here instead of a word like erinnert?

behält comes from behalten, which literally means 'to keep' or 'to retain'. While sich erinnern an means to actively recall something from the past, behalten is commonly used in learning contexts to mean successfully storing information in your memory so you do not forget it.

Why is man used twice in this sentence?

man is the impersonal pronoun meaning 'one' or the general 'you'. Because the indem clause is a full clause that requires its own subject to go with the verb trainiert, we have to repeat man to show who is doing the training.