One of the reasons, I suspect, that a lot of answers have been deleted is because the question itself isn't technically answerable.

Simply put: Rasputin did not cure Prince Alexei.

Longer answer: Rasputin was a peasant mystic, an archetype that was more common in Russia than we generally remember. He was a particularly entrancing one, but that was the limit of his quantifiable capacity.

He eventually met and gained the trust of the Tsarina because of his claimed ability to help her son, but there is absolutely no evidence he ever did anything to help him. Period.

People love Rasputin, so they love theorizing about him; mostly, these theories are needless. In general, Rasputin comforted the Tsarina, assuring her that the Prince would not die from various childhood hematomas. There's no evidence he gave the prince secret doses of aspirin, hypnotized him, or any of the other wild theories people like to put forward. On the most famous occasion, when the prince had a nearly-fatal injury on his leg, Rasputin was nearly 3000 miles away; his great act of healing was sending the Tsarina a telegram claiming her son would live.

Sure enough, the Tsarevich lived, but apparently Rasputin's healing was limited enough he still required therapy to correct the misshaping of his leg.

But the prince's hemophilia never went away. When the Bolsheviks finally executed him at age 14, he was in a wheelchair with yet another massive hematoma.

That's all very clear in any reading of the sources. But again, Rasputin is more a figure of pop culture than history at this point, and people write endlessly about him. You asked specifically if there are any theories on how he cured the prince: my answer is that he didn't, and there's no real way to read events in any way other than he was a lucky, charming charlatan with a good read on a woman desperate for good news about a beloved son.

That said, there are a number of "theories," some of which it appears people have put forwards. I'm not sure they're worth reading. Most are either inane or give too much credit to Rasputin.

Instead, not enough attention is paid to the enormous stress that her son's health put upon the Tsarina. She's often made out to be a dupe, a loon, or some other kind of hopeless idiot tricked by an obvious con-man. But her son, whom she loved deeply, spent the whole of his life on the cusp of death. He wasn't just 'sick;' he was so fragile that he required constant attendance from two sailors to protect him from injury. The slightest bruise could turn fatal. If he tripped going down the stairs, he would likely have died. He frequently had to be carried because merely the impact of walking took a deep toll on his legs.

For a woman with the wealth and power of the Tsarina, this was a nightmare scenario. She was an absolute monarch of one of the great states; her husband was dutiful and took pains to make sure she received anything she ever desired. But despite her enormous power, she had no ability to protect her son and the heir to the throne. The problem was ironically only compounded by the Tsarevich's alleged good cheer and likeable personality. He elicited great sympathy in anyone who met him.

Given the years she spent worrying over his constant struggle to survive, it's small wonder Rasputin gained the leverage he did. He didn't need to administer secret herbs or bribe the nurses to learn insider information. He was a charming, confident, entrancing man and he told a desperate person something they wanted to hear.

In other words, he didn't even really try to treat the prince at all: Rasputin was really more a therapist to the Tsarina.

Sources: So much garbage has been written about Rasputin. I'd ignore most anything that throws "Magic" or "Dark" or anything that sounds 'cool' out right off the bat. You lose some valid secondary sources but to be honest, you don't need them. It's a much simpler story than the cottage-indusry of Rasputin-writing makes it appear.

I'd rely primarily on the, uh, primary sources, many of which are collected here: http://www.alexanderpalace.org/palace/rasputinreport.html

By reading his letters, plus the secret police reports on his activities, you can really learn everything you need. He was just a charlatan who was lucky that the Prince did, in fact, survive the few years Rasputin was in court.

Hypnotherapy is a technique recognized by the ADA And the AMA as sometimes effective in stopping bleeding: when it works, it can help minimize pain and bleeding during dental surgery and procedures and has been effective under some situations in stopping hemorrhaging for accident victims. There are credible enough reports about it being used to lessen and relieve asthma attacks, migraines, certain GI problems and a host of other things. It just doesn't always work. Rasputin was not just seductive and enticing. The description of him as "hypnotic" is very likely factually accurate. If he hypnotized the tsarevich, and the tsarevich was indeed susceptible to that kind of influence, especially if it was practised in the context of religious faith practised by a Holy Man, then Rasputin's hypnosis might well have done the temporary trick to stop pain and bleeding.
No hocus pocus. But certainly not a "cure." Just enough to get Alexei through the worst of his bouts. Did Rasputin know this was the secret of success? Maybe: " mesmirism" had certainly been the rage in Europe for decades. But perhaps he came upon hypnotic ability naturally and saw it, whatever he called it, as his special gift from God.to the world.
If it helped seduce the Tsaritsa's friends, in the meantime, so much the better...