ELT stands for English Language Teaching. The teaching of English can be for it to be used as a second language, foreign language, international language, or for specific purposes.
In the field of English language instruction, it is imperative to have complete dominion of the language that is to be taught, because the goal is to prepare students to manipulate English as an L2, not only under a social and communicative context, but also for academic and professional purposes.
The connection between ELT and the study of Linguistics is that this is the field that scientifically breaks down a language system into its basic and most important components. Therefore, understanding a language from the "inside out" makes it much more likely that the instructor will be able to teach extensively, comprehensively and accurately.
Moreover, a strong knowledge of linguistics helps the EL teacher explain the language better to the students, since there is knowledge about its word formation (morphology), word meaning (semantics), the structure of words within sentences (syntax), and of the proper pronunciation and intonation of words in the target language.
ELT instructors who do not have a strong command of the principles of linguistics, nor its connection to the English language, will not give quality education to their students. Instead, they will be teaching students how to repeat words out of context and, perhaps, even pronounced erroneously. This is a current area of research in ELT considering that so many so-called language preparation programs are coming out but lacking fully-certified teachers, or teachers that have not trained to gain complete knowledge of the linguistic components of the language that they aim to teach.