Seemingly no division operation is done of the diagram of the transfer function Given the below system.
$$  y\left[n   \right] =2 x\left[n \right] -x\left[n -1\right] +0.5 y\left[n -2  \right]  $$
We'll find out the transfer function of it.
So, easiely using z-transform,
$$  Y(z) =2 X(z) -X(z) z^{-1} +0.5 Y(z) z^{-2}  $$
$$ Y(z) \left\{ 1-0.5 z^{-2}  \right\} =X(z) \left\{ 2-z^{-1}  \right\}   $$
$$ H\left( z  \right) = \frac{Y(z) }{X(z) }= \frac{2-z^{-1} }{1-0.5 z^{-2} }   $$
gained it.
Currently I can't get the below diagram.
I can't see component which operates division so it is weird that the diagram is applied to the transfer function.
About the blue domain of below, currently I can't get how the 1 of the denominator of the transfer function is attained.

What I've been missing or what should I study for next?
The diagram implements the time domain difference equation and NOT the frequency domain transfer function.
The difference equation has no divisions and therefore there are no divisions in the block diagram. The Z-transform is useful to analyze the frequency domain behavior of a system but you don't implement it directly. (at least not at this point in your learning).