Trouble understanding Grammar Rules
$(V\cup\Sigma)$ represents the set of variables (like $P$) and terminals (like $0$).

$(V\cup\Sigma)^*$ represents all finite strings (including the empty string) that can be made from variables and terminals (like $0P0$ or, $01P10PP0P0$, for instance). The expression $(V\cup\Sigma)^+$ is almost the same, but excludes the empty string.

Functions are essentially matchings so can be represented by a Cartesian product $(x, f(x))$, so in your definition, a production in $(V\cup\Sigma)^+\times(V\cup\Sigma)^*$ is a pairing from nonempty strings to strings, like the ones you gave. Others would include


$V\rightarrow \epsilon$ (the empty string is often denoted by $\epsilon$ or $\lambda$).
$S\rightarrow 01001$, or even
$aT\rightarrow TbT$