As the crew leave Planet Express at the end of the day, Fry and Leela pretend to go their separate ways, but after everyone else has left the locker room, the two emerge from hiding places and attempt to begin a romantic evening together.
However, they constantly find themselves being interrupted by the presence of Scruffy in the building, Bender unsuccessfully attempting to mug them during a walk in Central Park, Zoidberg as a busboy clearing their table in the middle of their meal at Elzar's restaurant, and Leela being self-conscious about Nibbler at her apartment.
The next day, the other crew members are loading the ship with tiny cars for a delivery to Simian 7, "the Planet of the Apes" where humans are strictly forbidden.
The point is driven home when they arrive at the human exhibit, where they find Fry and Leela enjoying their "secluded" vacation, unwittingly on public display on Simian 7.
The arrival of the tour group coincides with the moment at which Fry and Leela make up and begin their vacation in earnest.
Banjo helps them calm down by showing them a video of Fry and Leela complaining about and mocking each one of them in turn.
They emerge somewhat worse for the wear to find Fry and Leela just leaving, unaware of their presence, having spent a week on display.
When Fry and Leela loudly and mockingly revisit their own secrets about the others, Amy changes her mind and allows Bender to tell them the truth.
[1] Max Nicholson gave the episode a 7/10 "Good" rating, saying "Futurama's "Fry and Leela's Big Fling" featured a fun Planet of the Apes theme, but a not so great romantic arc.