Ibsen is often thought of as the grandpappy of realist drama. Other playwrights wrote in this genre—like Chekhov, Strindberg, Shaw, and O'Neill—but Ibsen was the pioneer.
In realist drama, the characters talk in a close approximation of everyday speech: no one is waxing poetic with "thees" and "thous" and no one interrupts a domestic moment to give a speech on, say, flowers. It's no big shocker that this trend stuck around. The vast majority of modern plays, TV shows, and movies are written in a similar style... though most fail to rise to the same level of social critique.
Check out this snippet of dialogue:
MRS. LINDE: "You must not forget that I had a helpless mother and two
little brothers. We couldn't wait for you, Nils; your prospects seemed
hopeless then." (3.22)
Straightforward? Oh yes. Conversational? Yeppers. Concerned with normal, everyday things, instead of, say, the fate of a nation? Yessirree Bob. This is some Ibsen-caliber realness being dished out.
A little lit history snack: realism shouldn't be confused with its cousin, Naturalism. Though the two styles were being developed around the same time, they have some big differences. Basically, Naturalism was just a lot more hardcore about representing everyday life exactly as it was: characters might talk on and on about nothing in particular and the plays might have no obvious climax. This could be a little on the yawn-worthy side, but hey: it's pretty dang natural.
Realism, however, is unafraid to be a little unrealistic. Look at A Doll's House. Sure the characters talk in a generally conversational way, but the plot is obviously and unapologetically contrived. There are melodramatic devices like top-secret letters. The doorbell rings at convenient times, bringing trouble for Nora. People enter and exit just when Ibsen needs to move on to the next scene and bring on new ideas.
This kind of staging wasn't a bad thing, in Ibsen's mind. His goal was to examine ideas and to challenge individuals to really think about their society... not to present photographic reality.