How should $\lambda x. \lambda y. f z x y \stackrel{\eta}{=} \lambda x. f z x$ be read?
Applications in Lambda calculus are assumed to be left associative, this means that $f~z~x~y$ is the same as $((f~(z))~(x))~(y)$.

With this is mind eta-conversion makes more sense. Wikipedia says:


  Eta-conversion expresses the idea of extensionality, which in this context is that two functions are the same if and only if they give the same result for all arguments. Eta-conversion converts between λx.(f x) and f whenever x does not appear free in f.


This means that $\lambda x. \lambda y. f~z~x~y \stackrel{\eta}{=} \lambda x. f~z~x$ is assumed to be true because $y$ does not change the result of the application $f~z~x$, its value is applied after $f~z~x$ is already evaluated.

You can find out more about eta-conversion by playing with it in the Haskell REPL:

Prelude&gt; let z = 1
Prelude&gt; let f x y = (+) ((+) z x) y
Prelude&gt; :t f
f :: Integer -&gt; Integer -&gt; Integer
Prelude&gt; f 1 1
3
Prelude&gt; let f x = (+) ((+) z x)
Prelude&gt; :t f
f :: Integer -&gt; Integer -&gt; Integer
Prelude&gt; f 1 1
3


As you can se, it is easily possible to leave out a the parameter y without changing the meaning of your program.