The speed at which Luke mastered the force is a testament to two main things. One being his sheer determination, which made him a quick study.
The second in his off the charts midi-chlorian count, being the same as that of Anakin. Luke was also in a prime setting on Dagobah, a planet Full of life and thus, full of the force. Also, he was training 12-14 hours a day for several weeks in a one on one setting with Yoda.
Don't forget that Luke already had some rudimentary abilities with the force due to Obi-Wan's introducing him to the force several years earlier. So that also helped him to bring his powers up to a far higher level in such a short time under the direct supervision of Yoda.
In the book "Shadows of the Empire" (SOTE), Luke was training in Obi-Wan's home on Tattooine for six months between the end of the original Star Wars "A New Hope" and the "Shadows of the Empire" book, which takes place between Empire and Jedi. He achieved a great level of skill and progress in that six months of non stop daily training and practice. So much so that by the time he went back to Yoda after rescuing Han in ROTJ, Yoda said "No more training to you require, already know you, that which you need"
During the SOTE, Luke was fighting Prince Xizor, head of the Black Sun galactic mob. He feared no one, was a master of martial arts and was not afraid of Vader either. He wanted to take him on directly, but said that the force make Vader too powerful to attack directly. He had a HRD (Human Relpica Droid) named Guri that was also his personal bodyguard. She was female and a machine, but was a lifelike as a terminator, with super strength, nearly invisible speed and amazing fighting skills. Luke had to take both her and Xizor on in direct combat, which he did and survived.
During the fight with GURI, who was undefeated until she ran into Luke, Luke attained a level of such oneness with the force, that even though she was attacking him at near invisible speed, each blow seemed to take several seconds or longer to reach him. Though she was giving him major problems until then, this helped him to get so deep into the force that he attained a new level of power in it. This is also one of the major incidents that helped him to grow so much stronger in the force between Empire and Jedi.
In my gong-fu training, I was also trained one on one with my teacher for about 3 hours a days 3-5 days a week. That plus I practiced at least 2 hours a day on my own, so that after only 3 months I was better than other students at the main school that had been training for 4 years, that according to my teacher himself.
So I know from personal experience that one on one training makes a HUGE difference in how fast and how well you learn. There is also the natural potential of the person. I've always been into the mind since I was 10 years old, practicing concentrating exercises and doing other mind training. So this also gave me the capacity to, at times, learn some techniques in a matter of minutes or seconds, to the shock of my teacher.
Luke would also have this similar ability due to his incredible natural force power and potential. This was confirmed when he was able to learn so much, so fast while training alone with Yoda on Dabobah as mentioned above.