The introduction of the three nameless boarders emphasizes how poorly Gregor is treated by his family. Prior to his transformation into an insect, Gregor lives an isolated life. He works long hours as a traveling salesman to pay off his parents' debt. He toils day in and day out, living a lonely, unhappy life, and is unable to form any meaningful relationships or establish roots because of the nature of his job. Meanwhile, his family enjoys the fruits of his labor. They have extravagant breakfasts, multiple newspaper subscriptions, jewelry, and a house staff. Gregor's family is not particularly loving toward him even before his transformation, and their treatment of him becomes outright abusive after his metamorphosis, particularly after the arrival of the lodgers.
After Gregor morphs into a bug, his father is cruel to him, threatens him with violence, and even attacks him. His mother and sister attempt to care for Gregor at first, bringing him food and clearing furniture out of his room so he will have more space to comfortably crawl and climb. Before long, these acts of kindness are replaced with revulsion and resentment. The Samsas are frustrated at having to work and support themselves now that Gregor is unable to financially provide for the family. They come to view him as a burden who has outlived his purpose. Their poor treatment of Gregor escalates after the three borders are introduced.
In an attempt to alleviate their financial distress, the Samsas rent out parts of their house to three tenants. The Samsas take great pains to ensure the comfort and contentment of their new guests, often at Gregor's expense. Gregor's family uses his room for storage in order to make space for the boarders. They dump unwanted and unneeded items in his room, which is symbolic, as Gregor himself has become unwanted and unneeded in their eyes.
The lodgers displace the Samsas and take over their house. They sit in the chairs that the Samsas used to sit in while being waited on and accommodated by the family, who are now servants in their own home: "They sat down at the head of the table, where in earlier days the mother, the father, and Gregor had eaten."
The Samsas neglect to bring food to Gregor but work hard cooking for the lodgers and are relieved when the food is to the tenants' liking: "He was satisfied, and mother and sister, who had looked on in suspense, began to breathe easily and to smile." While the lodgers are waited upon by the Samsas, Gregor is left without food: "How these lodgers stuff themselves, and I am dying." Prior to the arrival of the three guests, the door to Gregor's room was often left open. Now the door is almost always closed to keep Gregor's existence a secret.
While the Samsas dedicate great time and effort to the comfort of the lodgers, Gregor deteriorates and starves. He is left dirty, uncared for, and neglected:
because as a result of the dust which lay all over his room and flew around with the slightest movement, he was totally covered in dirt. On his back and his sides, he carted around with him dust, threads, hair, and remnants of food.
The introduction of the boarders emphasizes the Samsas' terrible treatment of Gregor through contrast.
We can also interpret the boarders as symbolic of the bourgeoisie. They move in to the Samsas' home, looking down on the Samsas as lower class (working class), and they expect the best of the food, the best of the rooms in the apartment, and so forth. They think very highly of themselves and feel very sophisticated with their long beards. Even when Grete prepares to play the violin for them, they rudely stand behind her, looking at her music over her shoulder, and then, when they realize that she is actually quite talented—something they did not expect from a working-class girl—they begin to talk to one another while she goes on playing.
If Gregor represented the working class initially, and his family the bourgeoisie (with their multiple newspapers and "lavish" breakfasts and lack of employment), once he becomes unable to work, they have to fill in and take on somewhat menial jobs themselves in order to pay the bills. When the boarders see Gregor, who has crept out of his room to listen to Grete play, they raise up a disgusted clamor, refusing to pay, and so forth. The family, however, led by Gregor's father, takes a stand against the boarders, actually throwing them out instead. We see, then, how music can inspire the proletariat class, the Samsas, to rise up and cast off the yoke placed on them by the bourgeoisie, the boarders. This is a very Marxist idea: that art can inspire and embolden the lower orders so that they may raise themselves up.
Franz Kafka's The Metamorphosis is composed of layer upon layer of symbolic meanings. Suffice it to bear in mind that the original German title -Die Verwandlung- admits of a number of translations, such as "change", "mutation", "conversion", and that it seems to be understood as "metamorphosis" only when associated with Greek mythology.
Why is this relevant to your question? Let's see. The three boarders behave as if they were a single character, although they follow the lead of "the middle lodger". During the Dorian (indo-European) invasion of Greece, that went under a different name at the time, a trinity of male gods under the leadership of one (Zeus) superseded the original female trinity -or three aspects of the Mother Goddess. Once the three boarders have settled down at the Samsa's, the three functional members of the family do their bid. They have, symbolically, been dispossessed of authority in their own home.
Gregor has been kept out of their sight, but one day he peeps out of his bedroom door, lured by the music played in the sitting room. The boarders are outraged by this disgusting creature and threaten to leave without paying their rent.
After this unsavory scene, Gregor dies in the small hours of the morning. His demise seems to bring his father back to his senses. He forces the boarders to look at Gregor's body and then throws them out of the house, thus regaining his authority and virility, for indeed it appears that the boarders have -again, symbolically- emasculated him.
Other interpretations claim that the boarders provide comic relief in an increasingly anguishing story. This could be justified by the following paragraph:
The gentlemen bent over the dishes set in front of them as if they wanted to test the food before eating it, and the gentleman in the middle, who seemed to count as an authority for the other two, did indeed cut off a piece of meat while it was still in its dish, clearly wishing to establish whether it was sufficiently cooked or whether it should be sent back to the kitchen.
In the multiple meanings that one could ascribe the boarders, an interesting one is that they stand for the harsh outside world, a world that will mercilessly crush the weak. That Kafka intended them to be symbols rather than characters is clear from the fact that he did not give them names. Moreover, the family's subservience to these one-in-three strangers for money may imply a warning against the lengths to which humans are prepared to go when moral values yield to selfish rationalizations.