Conditional Expectance ] Do you understand it ? And if you do so, can you please explain to me what this equality tells in simple english. If I understand the problem to prove then I can start working on it thanks.

$Y\mathbf1_{\\{Z=i\\}}$ stands for the product of random variable $Y$ and random variable $\mathbf1_{\\{Z=i\\}}$.

The second one takes value $1$ if $\omega\in\\{Z=i\\}$ (or equivalently if $Z(\omega)=i$) and takes value $0$ otherwise.

So $Y\mathbf1_{\\{Z=i\\}}$ is prescribed by $\omega\mapsto Y(\omega)$ if $Z(\omega)=i$ and $\omega\mapsto0$ otherwise.

This calculation of expectation of $Y$ must be made "under condition $Z=i$".

So we are simply not interested values taken by $Y$ for $\omega\
otin\\{Z=i\\}$.