"My friends stopped drinking."
Translation:Mis amigos dejaron de beber.
It would be a grammatically okay sentence, but it has a different meaning. Whenever you use a gerundio form, like bebiendo, it is talking about a progressive, ongoing action, i.e. you're not stopping. So you sentence would rather mean "We stopped in a drinking manner" or "We stopped while drinking".
In this case you don´t need to think how the sentence would be translated without de, it's just the way it is. In English when you say 'Someone + to be + going + to' you will never consider any other preposition except 'to'. The same thing is with the verb dejar, you just have to remember what preposition with what verb to use.
Some examples you can find here http://users.ipfw.edu/jehle/courses/VRBSPREP.HTM