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According to this lecture,"Radio astronomy has several advantages over optical astronomy...We can make the highest resolution images, and see things happening on the smallest scales."
Why is this? If resolving power, $r \propto \frac{\lambda}{B}$ then shouldn't shorter wavelength light result in smaller angular resolution i.e. better resolution?
The relation you cited holds for a single telescope. But, as also noted in the lecture you linked "One thing that is possible in radio astronomy is to use interferometry, which combines the signals from an array of antennas as if they were all part of the same aperture. That means that the resolving power of a radio telescope is not just what it would be for each antenna alone, but is as for a single telescope the size of the separation between antennas."
This basically means you use multiple telescopes - which in case of radio astronomy are called antennas for historical reasons - and combine their measurements into a single one. To get an intuition of how big of arrays we are talking about, google images of ALMA, LOFAR (which has stations in several countries) or SKA.
In principle other wavelengths, such as optical, can also be observed using interferometry, but it is technically much more complicated to build such instruments. On top of that, optical light is the one most affected by the atmosphere and it is hard enough to take it into account for one telescope, combining such effects to create an image from multiple ones is a very daunting task.
If you look at raw resolving power you get for a 10m optical telescope: $\theta_{optical}\approx \frac{5\times 10^{-7}}{10}\approx 5 \times 10^{-8}$ radian. For a synthetic aperture ground based radio telescope operating at 21cm the limiting aperture is of the order of the diameter of the Earth so: $\theta_{21cm}\approx \frac{21\times 10^{-2}}{12\times 10^{6}}\approx 1.75 \times 10^{-8}$ radian. So in this case the resolution of the radio telescope is somewhat better than that of the optical. In the absence of adaptive optics the situation is even more favorable to the radio telescope since we can achieve its theoretical resolution, but ground based optical telescopes are constrained to lower resolutions by atmospheric seeing. Also adaptive optics still do not improve the resolution of optical systems fully to the diffraction limit of the aperture.
