A simple way to see the use of a Field is to observe that in the generic communication design setting, we want to encode a k-bit string into an n-bit( n > k naturally) string so that the added redundancy can be used to combat the effect of noise. Any set of n-tuple can be thought of as a code. However a linear code facilitates easy encoding design and simpler decoding structures(though they may not be optimal) and hence we learn them in algebraic coding theory. A linear code is a vector space and vector spaces are built over Fields. Since most of the assumed source alphabet is finite, we talk about vector-spaces over finite fields, which are also termed Galois Fields.
It's not easy to explain the idea behind RS-encoding decoding or BCH decoding in a simple chat. Lots of results from finite fields are used to establish properties of BCH codes/RS codes and to facilitate their decoding.
So can you be more specific? Which part of BCH decoding have you not understood?
One of the, many ways to do BCH decoding, is to use the extended Euclidean Division Algorithm to find the error locator polynomial and error magnitude polynomial. Thus we find and correct errors.