A flat character is one that is only one dimensional or sometimes two dimensional. They only possess one character trait or possibly two. They can also be considered static characters because they don't change throughout the story; they remain the same. Flat characters are often stereotypical.
The characters Abram and Balthasar are flat because they are described as possessing very few character traits. In fact, we only see Abram speak in the first scene, and while Balthasar is with Abram, Balthasar does not have lines until later in the play. When Sampson and Gregory, Capulet's servants, insult Abram and Balthasar while they are on the street, asking them if they want to pick a quarrel, Abram says that he does not, in the line, "Quarrel, sir? No, sir" (I.i.53), proving that of the four servants, he is the one least likely to start a fight. However, when Sampson says that he is a "better" man than Abram is, meaning a better swordsman, Abram declares, "You lie," and begins the fight (61). Therefore, all we learn about Abram from this scene is that he does not want to start fights, but he will accept a challenge. He is no coward.
Similarly, we learn very little about Balthasar as well. Balthasar first speaks when he brings Romeo news in Mantua in Act 5, Scene 1. It is evident from this scene that Balthasar cares about Romeo's feelings, even though he is used as a comical character and delivers the news of Juliet's death with irony. When Romeo asks if Juliet is well and says that "nothing can be ill if she be well," Balthasar ironically replies, "Then she is well, and nothing can be ill" (V.i.16-17). The irony is that Juliet has been proclaimed to be dead and therefore can never suffer sickness or sorrow again. Therefore, in her state of death, Juliet is as well she will ever be again and therefore, ironically, according to Romeo, "nothing can be ill." We also see that Balthasar cares about Romeo's feelings and state of mind when he tells Romeo to calm down, as we see in the lines:
I do beseech you, sir, have patience.
Your looks are pale and wild and do import
Some misadventure. (27-28)
In the final scene, we also see that Balthasar cares about Romeo's state of mind when Romeo commands him to leave him alone in the tomb and Balthasar says in an aside that he'll hide and watch over things because he does not trust Romeo's intentions. Hence, we see that Balthasar is portrayed as a humorous and loyal servant. However, Shakespeare only allows us to see those two character traits and a flat character is one with only one or two character traits.
Hence, we see that because we are only given one or two character traits for both Abram and Balthasar, they are both flat characters.