The salamander used to be thought of as a legendary animal, able to withstand great heat and fire, and even set things on fire itself. While this is not true, the mythological aspect of the salamander has persisted in cultural thought. The salamander on Montag's arm is symbolic of his role as a "fireman," one who sets fires, and also of the "firetrucks," which in this culture are filled with kerosene to fuel fires.
...the firehouse stood all about in plaster silence and the orange Salamander slept with its kerosene in its belly and the firethrowers crossed upon its flanks...
(Bradbury, Fahrenheit 451, Google Books)
Because of the animal imagery that persists throughout the culture, the firetrucks are called "salamanders," and so Montag is a handler of the salamander. The patch on his arm shows that he sets fires and uses the Mechanical Salamander, the firetruck with its kerosene hose and flamethrower, to fulfil his job.