I love learning languages, and I can't imagine teaching without knowing at least a few words of the language of the students I am working with. The more of the language I have learnt, the more I feel it has helped my teaching.
Probably the biggest area it helps with is understanding the kind of grammatical areas in English that students with that L1 (first language) typically have trouble with. For example, Slavic language speakers change word order for emphasis in a way that isn't possible for English speakers as their case systems allow a flexibility which English doesn't have. The more Polish I have learnt, the easier I have found it to work out what students are trying to express in what can at times be quite complicated sentences in their writing. The tense and aspect system of Slavic languages also works differently to English: instead of distinctions between simple, perfect and continuous, Slavic languages have perfective and imperfective. Knowing this from having struggled with the distinctions in my own learning helped me to explain to an upper intermediate student why verb distinctions he felt should exist in English to describe the future were possible in Polish but not in English.
There are lots of vocabulary-related errors or concepts which my knowledge of the students' language has helped with. One interesting area is concepts which have a single word in one language, but two words in the other. These include pigeon and dove in English, which translate to a single word in Polish (gołąb), or cherries in English which can be czereśnia or wiśnia in Polish, depending on whether they are sweet or sour. Students who can't understand why we don't have two different words for cherries laugh when I try to get them to tell me the differences between what are quite clearly the same kind of bird!
Pronunciation patterns also differ between languages. There are sounds that it can be very challenging for non-native speakers to make without a lot of support, practice and/or concentration! Slavic speakers struggle to differentiate between /æ/ and /u/, for example cap and cup, but equally I can't produce what feels like many variants on English 'sh' and 'ch' when I speak Polish. Intonation patterns also vary, with Polish friends teling me that I sound very over-excited sometimes when I transfer English intonation to the Polish language. This helps me to appreciate how challenging it is for students to adopt the intonation patterns of a different language, and the way that English intonation can be interpreted by Polish speakers.
Of course, it's possible to learn about these L1 'transfer errors' without learning the language yourself, but I feel it's faster to pick them up along with the language. You can read about them, but until you are learning the students' language yourself, it's hard to really appreciate and internalise the differences between L1 and English.
What about in a multilingual classroom? I found that although my students were from many different countries, there were particular first languages which made regular appearances. For me, it was much easier to appreciate the challenges of students whose languages I spoke when they were learning English than those whose languages I knew nothing of. This encouraged me to try to learn bits of other languages to help me teach those students more effectively.
Being able to share your experiences of learning their language with students helps to build rapport, and means that I empathise with their struggles when learning English. Students also enjoy turning the tables and trying to correct my mispronunciations or teach me new words and phrases at break times or after lessons. We can explore both of our languages together, and in my opinion, this creates a richer classroom experience overall. Even if it's only a few words, I would strongly encourage you to try to learn at least a little of your students' language(s).