I am probably repeating was has already been posted, but here is the answer to your question, Rosieposie.
Anna Vyrubova, was the daughter of Alexander Taneyev, a popular composer of the time. He was also previously head of the Imperial Chancellery. Her mother was a Tolstoy. Her family was therefore part of Russian High Society. So it was natural that they would come into contact with the Royal family.They owned a country house on the Moskva River, quite close to Illinskoe, the country house of Grand DuKe Serge and his wife Elizabeth. She was the tsarina's sister.
She first met the Tsarina at her fathers house in 1901 I think, and during one of the Tsarina's subsequent visits later in the same year, for a second time.
This was during an epidemic of typhoid and Anna was in hospital. The Tsarina remembered her, and visited her.
When recovered, she was invited to the Alexander Palace for tea. Their friendship was cemented when Empress Alexandra discovered that she could play the piano very well.
After this visit Anna developed a sort of reverence for the Tsarina, and well, the rest is history.