Why does wind blowing against a strong current create way bigger waves?
There are two physical processes at play here:


Wind speed relative to the ocean surface
Wave focusing and blocking in opposing current


Wind speed relative to the ocean surface

As you describe in your question, wind speed relative to the ocean surface varies depending on the current orientation and magnitude. The rate of wave growth is proportional to the wind speed $U$ in direction of the wave relative to the phase speed of the wave $c_p$, squared:

$$
S_{in} \sim \left( U - c_p \right)^2
$$

However, the wave phase speed is modulated by the current:

$$
c_p = c_{p0} + u 
$$
where $c_{p0}$ is the intrinsic phase speed (phase speed in absence of currents) and $u$ is in the direction of the wave. Negative value of $u$ is then an opposing current. Wave growth is then:

$$
S_{in} \sim \left( U - c_{p0} - u \right)^2
$$

Let's plug in some numbers here. Let's say $U = 10\ m/s$, $c_{p0} = 3\ m/s$, and $u = \pm 1\ m/s$:


In the case of following current, $u = 1\ m/s$, $S_{in} \sim 36\ m^2/s^2$ ;
In the case of opposing current, $u = -1\ m/s$, $S_{in} \sim 64\ m^2/s^2$ ;


Thus, by changing the orientation of a moderate current in characteristic wind and wave conditions, we increased the rate of wave growth by a factor of 1.78! This effect will be more powerful the smaller the relative wind speed $U - c_{p0}$ is.

Wave focusing and blocking in opposing currents

Wave energy is advected (transported) by the waves' own group speed $c_{g0}$ and also by currents. If for simplocity we drop all the other terms in the wave energy balance equation, we get:

$$
\dfrac{\partial E}{\partial t} = -\dfrac{\partial \left[\left(c_{g0} + u \right)E\right]}{\partial x}
$$

The crux here is in the modulated group speed $c_{g0} + u$. If the waves propagate into an area of increasingly stronger opposing current, there will be a convergence of wave energy in that space, which will manifest itself in increase of high and steep waves. At the point where $c_{g0} = -u$, that is, the current is exactly countering the group speed of the waves, they can no longer propagate in that direction! Wave energy accumulates locally and vigorous wave breaking ensues.

Summary

Wave blocking is a different process from the wave growth dependence on relative wind speed described above, but, they often correlate because young windsea (short waves locally generated by wind) are almost always aligned with wind direction, so all you need is the currents to oppose wind for both of these effects to come into play at the same time. 

Both are likely to occur in strong and steady western boundary currents such as the Gulf Stream, Kuroshio, Aghulas, and Brazil current, and also in any region with strong tidal currents.

More on this topic in the answer to this question.