Let $M$ be a non-empty set whose elements are sets. What are $F={A×{A} : A⊆M, A≠∅}$ and $⋃F$?

$y$ is an element of $F$ iff it has the shape $y=Atimes{A}$ where $Asubseteq M$ and $Aneqvarnothing$.

The equality in your title already states this and cannot essentially be improved, so you do not have to bother about $F$ itself.

$x$ is an element of $bigcup F$ iff $xin y$ for some $yin F$.

Referring to the first line we get:

$x$ is an element of $bigcup F$ iff $xin Atimes{A}$ for some $A$ with $Asubseteq M$ and $Aneqvarnothing$.

That means that $x$ must have the shape $x=(a,A)$ where $ain A$.

Proved is now that:$$bigcup F={(a,A)mid ain Asubseteq M}$$ The condition $Aneqvarnothing$ does not need to be mentioned here because $ain A$ implies that $Aneqvarnothing$.