He had purposely stowed away during the transport of a tricesium warp core casing. He was found hiding in it by Seven of Nine and B'Elanna Torres. They noted he had a badly injured leg and brought him to sickbay.
The Doctor had difficulty treating him at first because he was extremely skittish and wouldn't allow The Doctor near him. The Doctor had discovered that he had an extremely large lung capacity, which allowed him to survive in an oxygen-less atmosphere for a certain period of time. He also seemed to have a voracious appetite. Seven of Nine seemed to take a liking to him and even gave him a portion of her rations to eat.
When Seven returned to see how he was doing, she found that The Doctor had managed to treat him when he seemed to calm a bit when The Doctor had begun humming the aria "Veglia, o donna, questo fiore" from the Italian opera Rigoletto. The Doctor had then decided to play the full orchestral version of the opera so he could treat him, which was successful.
The Doctor took him under his wing, and named him after the Phantom of the Opera, a character who likewise derived solace from music. The Doctor had discovered Fantome was very intelligent and had an extremely complex physiology. Fantome even had the ability to refract his life signs so that normal sensors couldn't detect him, which was how he had managed to stow away during the transport.
Seven then turned off the music, much to The Doctor's and Fantome's displeasure. She noted that perhaps Fantome couldn't speak but he could indeed hear. She began showing him objects and then linking them to a specific musical note to identify them. Fantome caught on to the idea in seconds and The Doctor continued to expand this system of musical notes as a language in which the crew could communicate with him.
It took The Doctor a while but he eventually managed to coax Fantome out of sickbay, though he was still very shy of nearly everyone on board. Tuvok encountered them on the way to the mess hall. He was pleased to see that Fantome was doing so well but Fantome quickly hid behind The Doctor, a little afraid of Tuvok. The Doctor reassured Fantome and told him that Tuvok was "the one who keeps us all safe." Fantome thanked him for this. Tuvok found The Doctor's method of communicating with Fantome very ingenious. Tuvok then wished Fantome well and after The Doctor had translated it for Fantome he apparently thanked him again and said goodbye.
In the mess hall Fantome was extremely curious of everything, especially all the different types of food. Bosaal, who Captain Janeway was negotiating with, became angered at Fantome's presence in the mess hall. He began berating the shy alien, who, sensing Bosaal's anger, immediately hid behind The Doctor. Bosaal called his species vermin and thieves. Bosaal even said that if he could find them on his ship he would kill them. The Doctor became angered at Bosaal's harsh words but Janeway managed to calm Bosaal down with the offer of scanning and taking on any member of Fantome's species that was on his ship.
In all Janeway brought three more members of Fantome's species on board. When she went to visit them in sickbay she heard the four communicate in Fantome's music language. She gave them an applause, thinking it was a musical composition until The Doctor explained that they weren't playing music but having a conversation.
Apparently Fantome had taught the other three how to use the music language in a mere fraction of the time it had taken him to learn.
He later assisted Voyager in its escape from the Void, along with the three other members of his species by sabotaging the engines of the ships that were attacking Voyager as it made its escape. (VOY: "The Void")