I always hear that the Sun will be one but never when we found that out or how.
In the 1940s and early 1950s the understanding of the evolution of stars was radically reformed. A key person was George Gamow, but others such as Edward Teller and Fred Hoyle and Geffory Keller also had a part.
It was discovered (Cecilia Payne 1925) that the stellar classifications (OBAFGKM) are a temperature scale. When stellar classification (and hence temperature) is plotted against luminosity one finds that there are some cool but luminous stars.
Since cool stars would give of less light per unit area, it follows that if they are highly luminous, they must have a huge surface area, and a huge volume. These are "red giants". Observations of the mass of these stars showed that they were very low in density. Gamow and Teller initially interpreted these as being young stars still in the process of collapse from a cloud of dust and being powered by cool cores of deuterium and lithium fusion. This was how Gamow described red giants in his 1940 popular science book.
But there are problems with this model. The distribution of red giants doesn't fit this model, one would expect such red giants to be found mostly in regions with lots of gas - ie nebulae, from which they could form. But most are not in such regions. Moreover the amount of deuterium and lithium in interstellar gas isn't enough to provide for the power output from a red giant, and more detailed modelling suggested that a "young red giant" model simply didn't work.
So between 1940 and 1950 the alternate model, that red giants were old stars that were "puffed-up" by hot cores of (degenerate) helium and powered by the triple-alpha fusion began to gain ground. When Gamow's book was reprinted in 1952, this new model was described. There were still uncertainties and it was not until the mid 1950s that a reasonably detailed mathematical description of the evolution from the main sequence to a red giant was complete.
So it was figured out using maths, and applying models of nuclear fusion, opacity, and the evolution of the chemical composition of the core of a star. And it was figured out by multiple people in the time span 1940-1955(ish)