“How Can You Teach Someone English If You Don’t Speak Their Language?”
This is a wide question of course, and may show a lack of familiarity on the asker’s part with what’s involved in learning a language.
I’d explain it like this: learning a language is a process that happens over time. It’s not that one day you don’t know it and the next you do, or that knowing a language is just remembering words; learners speak some English and then more and then more still. It’s possible to communicate in English without speaking it to perfection. My read on it is that modern ELT methodologies take it a step further and treat language not as something that you learns by having it explained to you in your own language, but as something you master by using.
More practically, many modern textbooks are designed to use only the target language. They are set up in such a way that simpler language and practical things are presented first, so while not every learner prefers it, it is in fact more or less possible for beginners to learn English in English, say, in a multi-lingual group with a teacher who does not speak their language.
Can things be sped up or facilitated with local language use? Sure, and as with using only the target language, there are both pros and cons to using L1. On of the advantages of using the target language only is that students are in a situation where they actually need to use the language to communicate, much like they will be in the real world. It also often goes hand in hand with utilizing deduction or other critical thinking skills to figure things out. This benefits learners not only in the long run (because they’ll use those skills in the future to be more independent learners), but in the present (because people learn and remember and just generally take in more when they are pressed to think about it themselves and not simply told the answer).