Why Abstract Algebra in Cryptography?
Abstract algebra basically comprises Groups, Rings, Fields, Vector Spaces, Modules, and many other algebraic structures. It is not only useful in Cryptography but in Channel Coding, in the branch of Chemistry and Physics too.
It is a tool like Groups where you can do arithmetic and algebraic operation under a closed set. One can compare two algebraic structures and find similarities between them and use these properties further.

Now your question about AES and Galois Field. if you do arithmetic operation on the matrix of AES, say multiplying two bytes $\texttt{0xEF}$ and $\texttt{0xDC}$, without finite field your many arithmetic operations go beyond 1 byte of length. Finite field make sure that it will remain within the closed set by reducing to modulo.

for example; if you take the finite field created by the prime number 11 ($\mathbf{Z}^*_{11}$) the set only contains $\{1,2,\ldots,10\}$ elements. All operations are done within this set.  Then only you can substitute bytes find inverses when you decrypt AES encrypted ciphertext.
if you don't use them, strictly speaking without irreducible polynomial (like in AES it is $x^8+x^4+x^3+x+1$) or prime number, where it makes sure that all length of all operations ends up in byte and hence easy to find their unique inverses while decryption.