How to find the intersection point in rootlocus diagram?
The characteristic equation is:

$$s(s+1)(s+3)=-K$$ 

Hence, plug in any point on the locus, and that will give the corresponding value of K.

It appears that your selected point on the locus is approximately: \$s=-0.4+j0.7\$, which will give \$K\approx 1.8\$

For an accurate answer to this particular problem (\$\small \zeta=0.5\$), write the CE as: $$s(s+1)(s+3)+K= (s+\alpha)(s^2+\omega _n s+ \omega_n^2)$$

solve for \$\alpha\$ and \$\omega_n\$, and hence find K. Thus:

$$ s^3+(\alpha +\omega_n)s^2+(\alpha\omega_n+\omega_n^2)s+\alpha\omega_n^2=s^3 +4s^2+3s+K$$
Giving \$\small K=1.83\$.

Systems higher than 3rd order will need a root solver.

But this approach begs the question, why bother to sketch the root locus in the first place?