General solution using Euclidean Algorithm  
I was able to come up with the integer solution that they also have in the textbook using the same method they used but I am really puzzled how they come up with a solution for all the possible integer combinations...how do they come up with that notation/equation that represents all the integer solutions ? I am talking about the very last line.

It's a special case of the following ubiquitous principle: the general solution of a nonhomogeneous linear equation is given by adding any particular solution to the general solution of the associated homogeneous equation. Namely, if $\rm\:L\:$ is a linear map then one easily proves
Lemma $\ \  $ If $\rm\ L\:v\ =\ n\ $ then $\rm\ L\:u\ =\ n\ \iff\ 0\ =\ L\:v - L\:u\ =\ L\:(v-u)$ 
Therefore $\rm\  L^{-1}(n)\ =\ v\ +\ ker\ L\ =\:\: $ particular + homogeneous solutions.
In your case $\rm\ L\:v\: =\: L(x,y)\: =\: 957\:x + 609\:y $ is linear, being the sum of linear functions.
A well-known example from calculus is when $\rm\: L = D\:$ is the derivative. The general solution of $\rm\:D\:f = g\:$ is the sum of a particular solution, i.e. an antiderivative $f = \int g,\:$ plus an integration constant, i.e. a solution of the homogeneous equation $\rm\:D\:f = 0\:$. For more see these answers.
