One of the most consistent character beats throughout Futurama is the sad state of Zoidberg's life, which is usually played for laughs. However, the sixth season of the series takes things in a very different direction when it reveals the tragic reason why Zoidberg is such an important figure in Farnsworth's life.
Let's look at why Zoidberg and Professor Farnsworth became friends, and how it explains the true bittersweet extent of the doctor's life.
Introduced in the second episode of Futurama, Doctor Zoidberg is nominally the medical staff for the Planet Express Delivery Company. While he's well-versed in various alien biologies from across the galaxy, he's constantly surprised and confused by human anatomy, which is complicated by the fact that almost all of his co-workers are humans. This has resulted in plenty of poor diagnoses and even dangerous operations, with Season 6's "Tip of the Zoidberg" showcasing many of his failed surgeries. This ends up being the breaking point for much of the crew, who confront Professor Farnsworth over Zoidberg's continued employment.
Although the crew isn't privy to the truth, flashbacks reveal that Farnsworth and Zoidberg have actually been friends for decades. During a period where Farnsworth was still willing to work with the nefarious Carol Miller -- aka the conniving and ultra-wealthy Mom -- Farnsworth and a group of soldiers were sent to Triton in search of a Yeti for experimentation. Zoidberg was the medical officer assigned to the mission, quickly befriending Farnsworth as the only two men of science present. When a poor landing resulted in the entire platoon being exposed to methane swamps on the moon's surface, however, all the soldiers are left bed-ridden with a deadly illness known as Hyper-Malaria.
Zoidberg and Farnsworth are the only two spared, and they work together to hunt down the Yeti, but Zoidberg suffers a serious head injury during the fight. Fortunately, Farnsworth saves Zoidberg during the battle by tackling the creature and killing it, prompting Zoidberg to take a vow at Farnsworth's request -- whenever the dormant Hyper-Malaria resurfaces in the Professor, the crustacean must be present to perform a mercy killing.
This results in Farnsworth hiring Zoidberg as one of his very first employees, and he remains a constant within the Planet Express. Decades later, the signs of the Hyper-Malaria resurface, so Zoidberg tries to help kill Farnsworth before he suffers a painful death. However, he discovers it's actually a sign that Farnsworth is being transformed into a yeti himself, an unknown side-effect of their fight with the creature. Zoidberg convinces Carol to give him the remains of the specimen so he can save Farnsworth, revealing that he turned down the chance to run his own lab for her and live a wealthy and comfortable life, all so he could repay his friend.
Zoidberg's destitute lifestyle was mined for plenty of jokes over the course of Futurama, with the character often bemoaning that he wasn't a successful figure. When he learned from Leela in Season 3's "The Cyber House Rules" that most doctors are actually quite wealthy, he reacted with shock and denial. But despite the opportunity to give himself a great life at any point, Zoidberg couldn't turn his back on his friend. It makes all of the comments and gags about Zoidberg's living conditions all the sadder, while also explaining why Farnsworth showed appreciation for the crustacean from time to time.
This backstory helps cement just how tragic Zoidberg is as a character. Episodes like Season 2's "Why Must I Be a Crustacean in Love?" and Season 4's "A Taste of Freedom" hint that Zoidberg has no real connections on his home-world, and his living conditions imply he has few friends outside of Planet Express, but despite all the abuse he suffers, he still sees them as his true friends and Earth as his real home.