Originally up in arms referred to weapons. It's not entirely clear as to why weapons were also called arms, however, it may be because many weapons, such as swords, clubs, daggers, and so on, are seen as an add-on to the arm.
Shakespeare used the phrase in his 1591 play, King Richard III. The line reads:
"March on, march on, since we are up in arms;
If not to fight with foreign enemies,
Yet to beat down these rebels here at home."
Throughout the years the phrase moved from being armed with weapons to being very angry or upset about something; having feelings of indignation; feeling infuriated at a person or thing. (source)