The field of Linguistics responds to need to study language within a scientific frame, including its core content (i.e. spelling, syntax, morphology, and semantics) as well as its social application in everyday dynamics... even in computational language!
This being said, there is no question that applied linguistics is one of the most useful areas within the field as it applies to English Language Teaching (ELT). The reasons for this pertain the application of linguistics in areas such as Speech Language Pathology (SLP), translation, reading, and Foreign Language (FL) Education.
The specific study of Applied Linguistics aims to explore how the compilation of codes, sounds and symbols that we call "language" affects everyday situations.
In the case of SLP, for example, applied linguistics are quite important because it is in the production of the correct sounds that we can create actual communication. For example, when school-age children show signs of poor speech, they are immediately referred for fear that these problems may affect their reading fluency and then, by default, their reading comprehension. This is why Applied Linguistics are important also in the area of reading. It all comes together.
Furthermore, through the study of applied linguistics we understand language in a graphic and pure way. Once you are able to break down a language into its main components, it is much easier to correlate the similarities, and understand the differences, between one linguistic root group and another. This is imperative in fields such as Foreign Language education,ELT, and in the study of language for purposes of translation.
Therefore, ELT, and Translation do not merely entail teaching or repeating words in an isolated context, far from it. These fields need a strong foundation in the understanding of languages so that they can then apply theories of motivation and learning in order to be successful with ELT. The field of linguistics that offers this type of training is applied linguistics. This is why it is a required course in most (respected and legit) ELT programs.