While the other Rugrats are off skating, and singing about how "Christmas-Channukah-Kwanzaa-and-Winter-Solstice-time is here", Tommy Pickles and Chuckie Finster walk over to the nearby fence, where Chuckie admits that he thinks he has a problem ("Diapie rash?" Tommy assumes). Chuckie goes on to quote Charlie Brown's speech from the beginning of the special being parodied, adding in mentions of Channukah, Kwanzaa, and Winter Solstice. He then proceeds to list activities associated with said holidays in addition to "sending cards and decorating trees", during which snow falls all around him and Tommy, covering them up entirely and eventually muffling his voice.
Later, in Tommy's playpen, the Rugrats are all dancing to a jazz-based rendition of "Jingle Bells", with Dil Pickles playing piano, and Spike on guitar. Eventually, Angelica Pickles (in the Lucy van Pelt role) calls a stop to the dancing to tell them all that Chuckie is coming with the decorations. During this, Spike mimics her movements behind her, and she makes a threatening glare in his direction. Chuckie comes in with a box containing a tree (just like the one Charlie Brown found), a menorah, and some corn, but he's still desperate for someone to tell him what the holidays are about. Tommy offers to do that himself, and proceeds to deliver the following speech in the manner of Linus van Pelt quoting the Bible:
"Lights, please. And there were, in the same country, people of different backgrounds, beliefs, and cultures. What ho! This shall be unto you a time of joy, peace on Earth, and goodwill toward peoples."
Angelica responds to this by screaming, "You blockhead! It's about the presents! Lots of presents!" The rest of the kids immediately agree upon this and all wish happy holidays to Chuckie. As the segment closes on the Rugrats all whistling "Hark, the Herald Angels Sing", an announcer tells us that the segment has been brought to us by Dork Krabby Mint Patties. Mr. Krabs then makes an appearance to endorse the advertised food.
A post on Manny Galan's blog, detailing the production of the Nickmas shorts, reveals that, for "A Chuckie Finster Christmas", they'd originally intended to have the Rugrats characters drawn and dressed to resemble the Peanuts characters they represented in the skit.  Mr. Krabs is pomoting Dork Kabby Mint Patties, a spoof of York Peppermint Pattes.