The fact that plants produce oxygen was discovered by Joseph Priestley in the 18th century (there were many other experiments by many other scientists to discover what oxygen is and its necessity for life; this answer will focus only on Priestley's experiment).

It was long known that combustion requires air. After all, a fire can be extinguished by smothering it. The mechanism by which this worked wasn't clearly understood; it was thought (incorrectly) that a substance called "phlogiston" was involved. Priestley set up an apparatus with a candle and a plant (mint, if I recall correctly) inside a sealed glass container. The candle quickly burned out and could not be relit. Others had done similar experiments and found that the gas left in the jar after the candle had burned out would kill a mouse that was put into the jar. Today we know that this was because the flame had used up all the available oxygen. But Priestley left the plant and the candle in the sealed jar for several days. He then found that the candle could burn again. Repeating the experiment demonstrated that a mouse could breathe in the jar after the plant had done its work. This demonstrated that a plant could "refresh" air that had been used up (or "phlogisticated," as the scientists of the day understood it). Additional experiments continued to refine our understanding of photosynthesis.