A black hole by definition does not emit radiation. In fact, it captures everything that falls onto it. So how do you observe something that does not radiate?

It is true that a black hole itself does not emit light. However, the EHT observes the nearby surroundings of a black hole. The gas that surrounds the observational targets of the EHT does in fact radiate, so by observing this region the EHT may observe structures that result from the strong gravity of the black hole.