There were several ways in which Dr. Roylott's Eastern training helped him in killing Julia Stoner and getting away with murder. The main way is referred to specifically in the text. Sherlock Holmes explains it to Dr. Watson after the case is resolved and they are traveling back to London the next day.
"The idea of using a form of poison which could not possibly be discovered by any chemical test was just such a one as would occur to a clever and ruthless man who had had an Eastern training. The rapidity with which such a poison would take effect would also, from his point of view, be an advantage. It would be a sharp-eyed coroner, indeed, who could distinguish the two little dark punctures which would show where the poison fangs had done their work."
The rapidity with which the snake's poison took effect prevented Julia from warning her sister Helen that she had been bitten by a snake and that their stepfather was responsible for sending it into her bedroom. Julia was delirious and could only mumble something about a "speckled band" and point to Roylott's room. Then she died.
"At first I thought that she had not recognised me, but as I bent over her she suddenly shrieked out in a voice which I shall never forget, ‘Oh, my God! Helen! It was the band! The speckled band!’ There was something else which she would fain have said, and she stabbed with her finger into the air in the direction of the doctor's room, but a fresh convulsion seized her and choked her words."
Roylott's Eastern training has also taught him how to handle poisonous snakes. No doubt he had spent time studying the cobra in India, but that snake would have been too big for his purposes. It could not have slipped through a hole in the ventilator, and furthermore it's venom is probably detectable in an autopsy. Roylott knew how to handle the Indian swamp adder safely. The only reason he was bitten and killed by his own snake was that he wasn't prepared for its sudden return after being beaten by Holmes' cane, nor was he expecting it to be in such an aggressive temper.
Finally, Dr. Roylott's Eastern training had taught him how to train his snake to return to his room when he blew a soft whistle and rewarded it with milk before locking it back up in his steel safe. It was the whistle that had warned Helen of potential danger. She remembered that her sister Julia had told her of hearing such a whistle for three nights in a row before the night on which she died. Helen only waited for daybreak and then went directly to London to ask Sherlock Holmes for advice and assistance.
During his summing up of the case, Holmes tells his good friend Watson:
When a doctor does go wrong he is the first of criminals. He has nerve and he has knowledge.