Spencer was born in Howland, Maine, orphaned at a young age, and raised by his aunt and uncle. He never graduated from grammar school, but went to work in a mill at age 12, before in 1912 joining the Navy to learn wireless telegraphy. He joined the Raytheon Company in the 1920s.
In 1941, magnetrons, which were used to generate the microwave radio signals that are the core mechanism of radar, were being made at the rate of 17 per day at Raytheon. While working there, Spencer developed a more efficient way to manufacture them, by punching out and soldering together magnetron parts, rather than using machined parts. His improvements were among those that increased magnetron production to 2,600 per day. For his work he was awarded the Distinguished Public Service Award by the US Navy.
In 1945, a peanut bar in his pocket melted while standing in front of an operating magnetron. He then tested popcorn in front of the magnetron (surely turning up the power and standing out of the beam), and it quickly popped all over the room. Development of the microwave oven grew out of these observations, and by 1947 a commercial oven was being sold by Raytheon.
He became Senior Vice President and a member of the Board of Directors at Raytheon. He received 150 patents during his career at Raytheon, and a building there is named after him.