There is a couple theories on Rasputin's "healing." Now, the majority of Rasputin's healing ability comes from him "healing" the young Tsarevich Alexei who had hemophilia which was transmitted through his bloodline.

I think first off that the idea that Rasputin was truly a mystic can be discredited right off.

Then we get to the other ones. There is one theory that gets repeated a lot, which I am unsure where the actual source is. The theory is that many European doctors, including the ones employed by the Tsar used aspirin as a magic cure-all for all sort of things at the time and Alexei was using aspirin when Rasputin came to the Winter Palace. Rasputin then swept away all "western" medicine including the aspirin. Now, this is important because aspirin (not known then) was a blood thinner. Blood thinner + hemophilia = bad time. This theory says that after Alexei was off the aspirin he felt better.

As I prefaced, this one is kind of all over and may be the "common consensus" but I really can't find the original source.

The other theory I've heard is Rasputin was calming to Alexei and lowered his blood pressure which made him feel better as well.

Edit: I've removed the section questioning on if Alexei really had hemophilia, which I dug up that he did. They found the bones and tested for hemophilia and found it present in both Alexei and his sister Anastasia. The source can be found from a BBC article: http://news.bbc.co.uk/2/hi/science/nature/8297467.stm