Why the negative reward function in linear quadratic regulation encourage to be at original state?
Minimizing $s^TUs=s^Ts=\|s\|^2$ minimizes the distance from the origin, because the norm of a vector is its Euclidean distance (https://en.wikipedia.org/wiki/Euclidean_distance) from the origin:

$$\|s\|= \sqrt{s_1^2+s_2^2+\dots+s_n^2} = \sqrt{(s_1-0)^2+(s_2-0)^2+\dots+(s_n-0)^2}$$