Rasputin had the ear of both the Tsar and Tsarina by convincing them he had magical healing abilities. We're the Russian royals just gullible idiots, and rasputin took advantage of that to gain power? Or did he believe he had powers and was insane?
Note to mods: This answer has been heavily re-worked from the previous one to most closely answer OP's question.
To start, this forum absolutely needs more answers on Rasputin. In the meantime, I'd suggest starting with r/kieslowskifan's answer to a similar question on Rasputin's fame.
Before we start: like all people, Rasputin did not exist in a vacuum - and was, as much as any person, a product of his time.
Of all figures in modern history, almost no other person seems as mysterious, esoteric, and all-out strange than Rasputin.
What’s more, while much has been written on Rasputin’s life, vast parts are shrouded in mystery and speculation.
Russian poet and diplomat Fyodor Tyutchev in 1866 famously wrote:
You cannot grasp Russia with your mind
Or judge her by any common measure,
Russia is one of a special kind –
In Russia, one can only believe.
Replace the word ‘Russia’ with the word ‘Rasputin’ and you’d go some way in trying to understand the man. Like most historical research, it’s best to attempt to understand the phenomenon of Rasputin with a very open mind.
In other words, we’re going to stick to the facts here, but it’s going to get weird.
Thankfully, the basic details of Rasputin’s life and time at the Russian court are well summarised at sources like Wikipedia. Thus, I’ll not go deeply into the ‘who was Rasputin and what did he do’ questions here.
To summarise the main events in Grigori Rasputin’s life:
With that very broad summation, we can now turn to Rasputin’s actions that won him such trust with the Empress Alexandra, and to a lesser extent, Emperor Nicholas II. (Note that Nicholas II technically was not a ‘tsar’ - but is commonly referred to as such, which I will do in this answer).
Let’s turn to Alexandra’s actual birth to Alexei, in 1904. Finally, a boy - and thus an heir to the dynasty! After the dynastically disappointing arrival of four daughters, previously, the imperial couple are elated. Yet sorrow very soon follows.
Incurable
The boy, Alexei, had hemophilia - an inherited disease which prevents or slows the ability of blood to clot.
While incurable, haemophilia - which generally only occurs in males - is not a death sentence. Yet living with the disease at that time meant avoiding any kind of injury, however slight.
Furthermore, the heir’s prospects of surviving any kind of future invasive surgery were also grim. As one of Alexei’s doctors warned: “The slightest bleeding may prove fatal.”
Now, faced with the prospect of appearing weak, Nicholas II and Alexandra endeavoured to keep Alexei’s disease a secret from the public, only revealing the heir’s poor health later on.
And although the state censor prohibited talk of Rasputin in the newspapers, naturally, rumours and Rasputin’s loud presence in St Petersburg would make up for their silence.
Casting a shadow
Writing later, the Imperial children’s French tutor, swissman Pierre Gilliard, summed up:
“The illness of the Czarevitch [Alexei] cast its shadow over the whole of the concluding period of the Czar Nicholas II.’s reign … for it made possible the phenomenon of Rasputin and resulted in the fatal isolation of the sovereigns who lived in a world apart, wholly absorbed in a tragic anxiety which had to be concealed from all eyes.” (note that for much of the tutor Gilliard’s time with the imperial family, even he did not know the secret of Alexei’s hemophilia.)
Let’s rewind to the weeks after Alexei’s initially joyous birth. Nicholas and Alexandra are devastated from his diagnosis of hemophilia, which ran in the family, and had already claimed the lives of Alexendra’s immediate relatives and close family.
No stranger to mysticism - and having previously turned to mystics in attempts to conceive a son and heir to the throne, Alexandra turns to the services of Grigori Rasputin.
Tall, unkempt, dirty, ill-spoken, and scandalously informal, Rasputin for the empress appears to perfectly fulfil a romanticised image of a peasant mystic - a noble savage free of the perceived worldly, gossipy corruption of the St Petersburg court elite the empress so despised.
At first, to the Imperial couple, Rasputin is something of an interesting subject - or ‘man of God’ as Nicholas calls him. The holy man does not on the first couple of meetings have on the agenda to heal anybody - and at this point has church clergy as his de facto handlers.
Then, in 1907, around a year after first meeting the imperial couple, returns to the palace, finding Alexei ill from a fall. The court doctors - of course some of the top physicians in Russia at the time - were powerless to aid the stricken boy.
Note at this point that hemophilia is not as well understand as it is today, meaning doctors may have had the tendency to try to do too much rather than let the disease run its course.
Alexandra sends for Rasputin to come to the palace - and in the morning, Alexei is a lot better.
Alexandra says that Rasputin prayed for the boy - and that fervent prayers to God were needed for such acts to take place.
The new go-to healer
This 1907 incident helped cement Rasputin as the couple’s go-to healer for Alexei. Contrary to what is commonly believed, Rasputin did not spend as much time with the Imperial couple as was believed. Their main residence, the Alexander Palace was on the outskirts far from central St Petersburg, where Rasputin was by this time living. Instead, the two would often communicate via letters.
While it was at this time still early days of Rasputin’s time at court, he seemed to know how his power was growing.
Yet even at this time, the imperial couple had evidence that Rasputin’s behaviour could be questionable.
Once, while on a palace visit, Rasputin asked the tsar’s sister Olga personal questions, and put his arm around her, stroking her shoulder. Olga jumped up to leave, while Nicholas and Alexandra looked uncomfortably on.
Meanwhile, back in St Petersburg, Rasputin became something of a spiritual guide to some women of the nobility, listening to their concerns and giving advice. The main scandals and significant events of Rasputin’s life were still to come.