In $t-t_0 = \pm \int_{x_0}^x \frac{dx'}{\sqrt{\frac{2}{m}(E-U(x'))}}$, how do you get $x(t)$? Why does integrating a function of $x$ result in a time difference?

See above question. You plug in $U(x)$ to get some function of $x$ on the RHS, then solve it to get $x$ as a function of $t$. The time difference comes from integrating $dt$ in the original differential equation.