English kept the end of the word and dropped the person. Rickshaw is what is left of jinrikisha, and the part cut off is jin, "man."
Etymonline breaks the Japanese word into three pieces: jin, "a man," riki, "power," and sha, "carriage." Wikipedia renders the whole as "human-powered vehicle." Etymonline adds that the elements are said to come ultimately from Chinese.
The full form reached English first. Etymonline dates jinrikisha to 1873, in English-language publications in Japan, where the vehicle was still new. The short rickshaw follows in 1885. Rudyard Kipling helped spread it with a ghost story, spelling it 'rickshaw, with an apostrophe at the front.
The vehicle itself was a recent invention. Wikipedia says rickshaws were invented in Japan around 1869, after a ban on wheeled vehicles was lifted. Izumi Yosuke is described as the most likely inventor, working with Suzuki Tokujiro and Takayama Kosuke. From 1870 the Tokyo government gave the three men a permit to build and sell them. Anyone running a rickshaw in Tokyo needed their seal.
The idea spread fast. By 1872 about 40,000 rickshaws were in service, according to Wikipedia. They replaced the older kago and norimono and became the main mode of transport in Japan.
Other inventors have been proposed. An American missionary, Jonathan Goble, is said to have built one around 1869 for his invalid wife in Yokohama. Claims also exist for an American blacksmith, Albert Tolman, in 1846, and for a carriage maker, James Birch, in 1867. Wikipedia still names Izumi as the most likely inventor.
From Japan the rickshaw moved across Asia. Wikipedia says it was first seen in China in 1873, with 10,000 in operation within a year. It reached India around 1880, starting in Simla.
The name outlived the man pulling it. Pedal-driven cycle rickshaws and motorized auto rickshaws kept the word. Most Indian cities offer auto rickshaw service, and Sri Lanka had over a million registered by 2018.