I'm currently watching Scoratica's YouTube series on Group Theory. On one of the videos, the following argument is made ($N$ is a normal subgroup of $G$): Condition for the cosets to act like a group
I don't understand why $xy \in (xN)(yN)$ is a necessary condition. We haven't even defined what $(xN)(yN)$ means yet. It could be the set formed by multiplying every element in $xN$ by each element of $yN$, it could be the set formed by the union of the sets $xN$ and $yN$, etc. Is there only one definition of the operation $(xN)(yN)$ that allows the cosets to behave like a group? If so why does it include $xy$?
There are two natural approaches to studying groups (and algebraic objects more generally). The first is to study subgroups, and the second is to study their operation-preserving maps (homomorphisms).
Quotients arise naturally from the study of homomorphisms, in that the cosets are the non-empty preimages of the homomorphism. That is, let $\varphi : G \to H$ be a homomorphism. Now let: $\{ \varphi^{-1}(h) : |\varphi^{-1}(h)| > 0 \}$ be the set of preimages that are non-empty.
Take $x \in \varphi^{-1}(h_{1}), y \in \varphi^{-1}(h_{2})$. So $\varphi(xy) = \varphi(x)\varphi(y) = h_{1}h_{2}$.
Note that if $N = \text{ker}(\varphi)$, then $x \in xN, y \in yN, xy \in xyN = xN \cdot yN$.
There are technical tools that need to be developed to equate these concepts, but this is the high level idea as to why cosets appear naturally and why the operation $xN \cdot yN = xyN$ is the correct operation.