Why was "the Santa Claus story" included in Gremlins?

It enforces the film's mix of Yuletide horror and comedy, and also fleshes out Kate's character a bit more.

You're not the only one who felt this story was out of place: this scene wasn't initially a big hit with the film's executives, either. From MentalFloss:

When the rough cut was complete, both Spielberg and some Warner Bros. executives wanted [the story] cut, as it wasn’t clear whether it was meant to be sad or funny. Dante insisted that that’s what made it a perfect metaphor for the film itself, and insisted it be kept in. In Roger Ebert’s three-star review of the film, he singled out this scene in particular, citing her story as being “in the great tradition of 1950s sick jokes.”

But it's worth remembering that Kate's 'real life' story isn't original - she's actually repeating a well known Xmas urban legend. And those who are familiar with it (and with Dante's sense of humor) might already sense the right tone: Urban legends and campfire stories like these aren't meant to be deep or tragic but rather morbid and fun. Dante himself has elaborated on the right nature of this scene in this interview to Film Comment, adding that it was also meant to supply a bit of background for Kate's character (the monologue wasn't originally written for her):

The tone of the movie is summed up in that speech. She’s telling a story that’s completely ridiculous. However, if it actually happened to you, it would be horrible! It’s like the guy falling on a banana peel. It’s funny if you’re across the street, but not funny if you break your back. I like the complexity of it. Originally, it wasn’t her character who told that story, it was a guy who owned a McDonald’s. At the McDonald’s the gremlins would come in and eat the people but not the food. When that character and that bit disappeared, I said to Chris, I hate to lose that character and that speech. Let’s give it to Phoebe’s character, Kate, because she doesn’t have much stuff going on except being the heroine, and this gives her a secret. The audience has to find out what it is. She doesn’t like Christmas—well, why not? And now we find out why. And that’s a better character arc for her. And she did it beautifully.

Because it's funny:

Here’s the thing, though: it’s funny. Gut-bustingly, brutally, aggressively funny. I’d always found it funny on my own; everything about the way it’s framed hints that despite the tone it’s not intended to be taken seriously. The fact that it cuts away to Gizmo at all is hilarious. He’s a ball of fuzz that can’t get wet or go in sunlight, the Earth’s most abundant resources. Who the hell cares what the fuzzball is doing? But there’s also the reactions from everyman Zack Galligan, who’s surrounded by an enormous festive silhouette reading, “Christmas!” It’s patently farcical and utterly human at the same time, playing the scene as absurdist but also satirizing the thick, saccharine pap that typifies most Christmas movies. Bad things happen at Christmas too; bad things are happening just outside the bank where Cates and Galligan are hiding. When Christmas sucks for you, no matter the cause, it sucks hard.

It’s hard to describe precisely how well this scene plays in a packed theater. It comes almost immediately after the movie’s most memorable effects blitzes, when Cates is serving dozens of gremlins at the bar across the street. Dante is almost manic in his depiction of gremlins indulging in as many vices as possible: gremlins drinking, smoking, flashing in a long trench coat when they have nothing to flash, gunning each other down, rocking ski masks and attempting old fashioned ‘80s muggings, breakdancing, gambling. Name something your parents told you not to do, and the gremlins are doing it in the bar. When it finally ends, the crowd in the theater is dizzy with ascending laughs, and then Dante gives them just enough time to breathe as Cates and Galligan escape and look for somewhere to hide. Then he hits them with the Worst Christmas Story Ever Told. And he knocks them down all over again.

Also:

It’s honestly shocking that a mainstream movie has such an uncompromising, bleak exchange. This is the part they try and leave out of kids’ books, but it’s also the engine that drives this occasionally cruel, frequently hilarious, and bizarrely heartfelt film. Christmas is a wonderful time of year, except when it isn’t, and when it isn’t, it’s the worst. Kate’s dad didn’t just die; he died stuck in a chimney, dressed like Santa Claus. That’s not a tragedy; that’s a fucking novelty song.

But he’s still dead, and Kate is still fatherless. And the Gremlins are still tearing up the town. It’s only funny when it’s not your home being destroyed, when it’s not your loved one rotting just a few feet away. It’s only merry when you’re happy and safe and get to spend the holidays with the people you care about. If it’s December 24 and your parents are gone, and your siblings won’t speak to you; if you’re getting older, and your future stretches out like a wasteland of broken promise and barren ambition; if you know that the next time you hear “Wonderful Christmastime,” you might finally snap and crash your junk heap of a car into the nearest Salvation Army Santa, then all of those holiday greetings can sound like the shrieks of the mocking damned.