How is Six Characters in Search of an Author an example of metatheater, and what are its metatheatrical aspects?

The term "metatheater" refers to drama which is self-referential and makes references to its own theatricality. In other words, in "realistic" theater, actors on stage pretend to be the characters in the play and do not "break the fourth wall" by referring to the fact that they are actually actors or talking directly to the audience. Instead, they act as though they are simply people living in a world with a transparent "fourth wall" through which the audience can view them. Metatheater in the western tradition begins with the "parabasis" of Greek comedy, in which an actor addresses the audience asking for the audience to vote in favor of the play winning a prize at a festival. Since antiquity, comedy has tended to retain a metatheatric component, with actors making asides that directly address the audience for humorous effect.
The plot of Pirandello's play is metatheatric in that it is entirely about how plays are made, written, and rehearsed. The "characters" are constantly talking about themselves as dramatic characters, and the process of crafting and rehearsing plays is the main subject of the plot and dialogue. There is no attempt at a realistic illusion of a fourth wall, but instead the audience is constantly reminded of the artificial and artistic nature of the theater. Even "improvisation" is shown as something self-consciously crafted rather than natural and spontaneous.

Metatheatre is a type of a play (or a part of a play) that exposes the normally hidden aspects of theater, most importantly the distinction between fiction and reality. A regular performance in a theater does everything to make the audience forget that they're looking at a play. Modernist plays experiment with that idea: for example, how many props and other visual tools can you take away to still allow the audience to "see" what's supposed to be on stage.
Metatheatre goes a step further and straight-out draws attention to it, which is the reason why a common metatheatrical element is play-within-a-play, as it is with Six Characters in Search of an Author. Pirandello's work is one of the best examples of metatheatre, since the premise itself is metatheatrical. The play begins with a director conducting a rehearsal, which is interrupted by six Characters who want to have their story told and finished. The fact that the Characters are "intruders" in a way is only the first element, but it does set the tone. The Characters are sort of lost, they're not supposed to be a part of the story. The audience gets the feeling that they're watching something that's happening behind the curtains.
Pirandello explores the metatheatrical aspects further, however. The Director agrees to let the Characters solve their tale, but they are not happy. First, they're not pleased with the setting and the stage, saying it's not realistic and truthful. Secondly, they can't stop laughing when the Director tries to make his Actors play them. Everything is wrong: from the accents to their movements. Again, light is shown on something that usually takes place off-stage and long before the play is ever seen.
Reality and fiction fuse together the most at the end of the play where the Characters resolve some part of their story line on stage, continuing the re-enactment with real-time interaction. They move smoothly from putting on a play about themselves to actually being themselves—only on a metatheatrical level they're still on stage, from the point of view of the real audience watching Pirandello's play. It's these types of meta-fictional moves and elements that make Six Characters in Search of an Author a very prominent metatheatrical work. The play constantly blurs the lines between acting, reality, and some kind of mixture of both.