The distributive law $4\left (x+y \right)=4x+4y $ because $4\left (x+y \right) =\left (x+y \right) +\left (x+y \right) +\left (x+y \right) +\left (x+y \right)$ , but why is $\left (x+y \right) \left (x+y \right) =xx+xy+yx+yy$?
 A: Because:
$$\left (x+y \right) \left (x+y \right) = \underbrace{(x+y)+(x+y)+...(x+y)}_{x + y\text{ times}}$$ 
$$=\underbrace{(x+y)+(x+y)+...(x+y)}_{x\text{ times}}+\underbrace{(x+y)+(x+y)+...(x+y)}_{y\text{ times}}$$
$$=\underbrace{x}_{x\text{ times}}+\underbrace{y}_{x\text{ times}}+\underbrace{x}_{y\text{ times}}+\underbrace{y}_{y\text{ times}}$$
$$=xx+xy+yx+yy$$
A: 
$4\left (x+y \right)=4x+4y $ because $4\left (x+y \right) =\left (x+y \right) +\left (x+y \right) +\left (x+y \right) +\left (x+y \right)$

This works because $4$ is a (positive) integer; but for non-integer factors you can't use the "repeated addition"-argument.

but why is $\left (x+y \right) \left (x+y \right) =xx+xy+yx+yy$?

The distributive law works even if $x$ and/or $y$ are not integers.
Keep one of the factors together in a first step, and apply distributivity twice:
$$\begin{align}
(\color{blue}{x}+\color{red}{y})(x+y) & =\color{blue}{x}(x+y)+\color{red}{y}(x+y) \\ & =\color{blue}{x}x+\color{blue}{x}y+\color{red}{y}x+\color{red}{y}y
\end{align}$$
A: Do it in steps.  
You accept that $M(x+y) = Mx + My$
So replace $M$ with $(x+y)$ and you get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y$
Now distribute a second time:  Replace $x$ with $A$ and $y$ with $B$ to get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y=$
$(x+y)A + (x+y)B=$
$xA + yA + xB + yB =$
$xx + yx + xy + yy =$
$x^2 + 2xy + y^2$.
Of you you don't have to, and you shouldn't, do all that replacement.  You should do it directly. 
$(x+y)(x+y)=$ we treat one of the $(x+y)$ as a single thing and distribute across the other $x + y$.
$(x+y)(x+y) = (x+y)x + (x+y)y=$.
Now we have to sums to distribute: $(x+y)x = xx + yx$  and $(x + y)y = xy + yy$.  So puting them together:
$(x+y)(x+y) = (x+y)x + (x+y)y=xx + yx + xy + yy=$.
And then some clean up:
$=x^2 + 2xy + y^2$
