why does the off diognal elements of the matrix mediate with the coupling differential equation?
The off-diagonal elements cause the second derivative of $x_1$ to depend on $x_2$ and vice versa. That’s what it means to have coupled differential equations.
If the off-diagonal elements are zero, then you have two uncoupled equations: one for $x_1$ and one for $x_2$.
Uncoupled equations are easier to solve because you can solve each one separately from the other.