Yet how can he get tested with no symptoms and no history of exposure?
He was exposed. He attended a social event in Louisville, KY (the state he is a senator for), and several of the attendees have tested positive. The proper thing to do is test everyone who was at that event. That level of testing doesn't happen yet for everyone in the US, because there are not enough tests to go around. But we are at the point of having done 8 tests for every positive result, so we are no longer just testing people with symptoms, like it was at first.
Two things will stop this disease: cutting physical interaction as much as possible, and aggressive testing and tracing of people when one tests positive.