Linear Approximation Question (Solution included)

I don't understand how to do this at all and I don't get why the solutions do what they do.

1) Why do we need the partial derivatives with respect to x and y?

2) What's going on with the f(x+h,y+k)? It must be some kind of a definition. It seems familiar from the limit definition of a derivative but I can't connect the dots with what we're using it for if that's what it is.

3) I also do not understand the fundamental idea of finding a linear approximation at (2,1) for approximating f(1.95, 1.08).

I REALLY need help and any would be greatly appreciated!
Thanks in advance!