Waiting for Godot and The Birthday Party are examples of postmodern literature. eNotes's Guide to Literary Terms offers an ample definition of postmodernism, and one of the most salient details is that
the authors abandon the concept of an ordered universe, linear narratives, and traditional forms to suggest the malleability of truth and question the nature of reality itself
This is essentially the way that fragmentation is used in the two plays. The dialogue is broken into pieces that are immediately followed by pauses. This defies the idea of linear conversations in which actions have reactions, reasons, and consequences.
Having a traditional dialogue in either of these plays would suppose that art is imitating reality. However, in Waiting for Godot, as well as in The Birthday Party, art has no intention to imitate reality in any way.
Another aspect of postmodernism that is evident in the use of fragmentation is that
language is inherently unable to convey any semblance of the external world, and that verbal communication is more an act of conflict than an expression of rational meaning. (eNotes's Guide to Literary Terms)
Therefore, fragmentation is presented in the way the dialogues are written, because they are dispersed, followed by long pauses, and not necessarily substantial:
MEG. What are the cornflakes like, Stan?
MEG. Those flakes? Those lovely flakes? You're a liar, a little liar. They're refreshing. It says so. For people when they get up late.
STANLEY. The milk's off.
MEG. It's not. Petey ate his, didn't you, Petey?
PETEY. That's right.
MEG. There you are then.
STANLEY. (Pushes away his plate.) All right, I'll go on to the second course.
MEG. He hasn't finished the first course and he wants to go on to the second course!
A dialogue of this kind will not lead anywhere, is entirely ornamental, and does not serve to advance the plot. It is not a pivotal dialogue. It just exists there for that moment in the play. There are examples of this kind of fragmented dialogue in Waiting for Godot as well:
ESTRAGON. What did we do yesterday?
VLADIMIR. What did we do yesterday?
VLADIMIR. Why ... (Angrily.) Nothing is certain when you're about.
ESTRAGON. In my opinion we were here.
VLADIMIR. (looking round) You recognize the place?
ESTRAGON. I didn't say that.
ESTRAGON. That makes no difference.
Another example of the use of fragmentation is the way in which the characters are unreliable since their own reality is fragmented. This is all part of the genre's effort to break established norms. Both plays have characters who seem to be either clueless or lost in their own reality. This is also a way to reject reality as it is and show it represented in art. In this way, then, the fragmentation seen in dialogue in Waiting for Godot and The Birthday Party contributes to the postmodern style of the plays.