Why is $x^2+x$ the same as writing $(x+0.5)^2-0.25$? I find it extremely weird that a translation vertically by a variable should cause the graph to move both vertically and horizontally. Also, why 0.5 and 0.25?

We can use the completing the square method on $x^2+x$ to achieve this result:

Given an equation of the form $x^2+ax+b$ we can write this as $$\left(x+\frac a2\right)^2-\left(\frac a2\right)^2+b$$

So, in this scenario, we have $a=1$ and $b=0$, so we have \begin{align}x^2+x&=\left(x+\frac 12\right)^2-\left(\frac 12\right)^2+0\\
&=(x+0.5)^2-\frac 14\\
&=(x+0.5)^2-0.25\end{align}
