But that explanation is lost to time.

One translation I read suggested a probable explanation.

Rasputin’s phone advice was the same as many modern quacks: keep the patient away from modern medicine and doctors.

So the hemophiliac prince was no longer given his normal cocktail of drugs, which probably included a new medicine for the time: aspirin.

Stop giving a blood thinner to a hemophiliac and his condition (temporarily) improved. The best explanation for the people at the time was “magic”.