When did film actors begin acting more realistically?

Acting historian here and apologies for mobile. The short answer is about 1931 in theatre and 1951 in film (give or take), thanks to the popularization of Lee Strasberg’s adaptation (followed by Stella Adler’s and Sanford Meisner’s competing versions) of Konstantin Stanislavsky’s approach to acting. The three (along with Uta Hagen, who came later) are popularly and historically (though in my opinion erroneously) lumped together as “the Method” or “Method” acting.

Longer answer:

I cannot grant the premise that acting got fundamentally more realistic in these eras. The vast majority of advances in European acting styles/theories from about Shakespeare on were praised for their increased sense of naturalism. In some cases, the “nature” of one generation becomes the stilted fakery overthrown by the next generation. Instead, our sense of the natural changes as we evolve and change.
In the case of the so-called “Method,” it gained popularity in New York when the Group Theatre (including Strasberg, Adler and Meisner) premiered its first show in 1931. They were a scrappy young company that put a lot of work into ensemble acting instead of putting a star center stage and treating everyone else as a supporting player. Why did they hit it big? Most historians of the era point to the revolutionary politics of the Depression and a renewed interest in art about poor people instead of the wealthy. Group Member Clifford Odets quirky turned from acting to playwriting and his play Waiting for Lefty still stands as a landmark piece of Labor Theatre. Their gritty portrayals of the working class proved very popular and their approach to acting caught on like wildfire even if their own artistic infighting and unsustainable economic model meant that the Group itself dissolved entirely within a few years. Group members went on to successful careers in theatre, and many of them went on to become the most influential acting teachers of the Twentieth Century.
I choose 1951 for film because it was the premiere of A Streetcar Named Desire with Marlon Brando. (It’s worth noting that Vivian Leigh is the putative lead in the film and was herself part of the latest “realism” movement in her native England even though her work in no way resembles Brando’s.)

Brando’s performances in that and some subsequent films were completely uninterested in the kind of polish people expected in their films. In previous decades, even when actors were playing “average” people - think Henry Fonda or Jimmy Stewart - they were still careful that every word and motion was clearly visible to camera and audience. Brando - himself a student of Stella Adler - seemed totally uninterested in the camera and gave this sense that he was just existing instead of performing. The raw emotion and animal-like quality he brought to the screen was electrifying and made others look like they were working too hard in their diction.

There have been a ton of backlashes against the so-called “Method,” and I would be wrong to give you the impression that every actor since has used any version of this or that it is a singular practice. Instead I would say that our tastes in acting have changed over time and will continue to change. Tom Hanks is a fine actor and certainly my could be called “realistic,” but looks very little like Brando. Meryl Streep is a genius and trained in this school of acting, but her work is all her own. Will Ferrell has had a long career and no one would say that he is primarily known for being “realistic.”