Suppose we have a system $G(s) = \frac{1}{s(s+1)}$ and controller $K$ (this is purely a gain) and we close the loop:
$$T(s) = \frac{KG(s)}{1+KG(s)} = \frac{K\frac{1}{s(s+1)}}{1+K\frac{1}{s(s+1)}}$$
$$ = \frac{K}{s^2+s+K} $$
As you might notice, the poles of this closed-loop equation depend on the value of $K$:
$$s^2 + s + K = 0 \rightarrow s = -0.5\pm\sqrt{0.25 - K}$$
This means that one can influence the behaviour by only changing this $K$ to any arbitrary real value (imaginary might sound cool in simulation, but its kinda hard to put an imaginary voltage to a system for instance). The root locus plot represents for how the poles shift for changing values of $K$ (where $K > 0$). As you stated, if $K$ can only be an integer, you will indeed not get a continuous function, only dots at the places where $K$ exists. But as I stated earlier $K$ can by any real value, however for negative values you can quickly see the system becomes unstable.
EDIT: I have rewritten this part in a more elaborate proof.
Instead of looking at a specific system, suppose an arbitrary system
$$G(s) = \frac{N(s)}{D(s)}$$
The angle condition is the point at which the phase of the open loop system is an odd multiple of $-180^o$ or in other words:
$$\mathcal{Im}\left\{\frac{KN(s)}{D(s)}\right\} = 0 ~~\text{ and }~~ \mathcal{Re}\left\{\frac{KN(s)}{D(s)}\right\} < 0$$
The magnitude condition is the point where the magnitude of the open loop transfer equals 1. If both the magnitude condition and the angle condition match, the denominator of the closed loop transfer function becomes 0 (which is something we tend to avoid at all cost). Now, suppose we take a value for $K$ and calculate the value for $s = s_0$ such that the open loop transfer equals $-1$:
$$\frac{KN(s_0)}{D(s_0)} = -1 \rightarrow \frac{N(s_0)}{D(s_0)} = -\frac{1}{K}$$
We know this $s_0$ lies on the root locus. What you might also notice is that the sign does not change if I change $K$ to any positive, real value. Therefore, if $s$ lies on the root locus, the angle condition holds for any positive, real $K$. You could also reflect this in the bode plot, as $K$ only changes the magnitude, not the phase plot. As you might expect, this also implies the inverse holds: the angle condition holds for any value of $s$ that lies on the root locus of $G(s)$.
The magnitude condition only holds for a finite set of values of $s$ on the root locus (the solution at which the characteristic equation equals 0).
I hope I explained it a bit better, you can try also to just write out a couple cases and you quickly notice the same.
