Glad you asked... allow me to explain.
The rules of pairing beer with food are actually pretty similar to pairing wine with food. The fun part though is that beer offers such a broader range.
1. Like cancels like.
What this essentially means is that like flavors don't enhance each other. In fact it's the exact opposite. Like components between a beer and a dish will tone each other down, which then allows you to taste more of the other flavors in the beer and dish. The dish bellow is an open faced smoked salmon "BLT". The sauce on top is an aioli made with Jester King Noble King. That was paired with the Jester King Farmhouse Wytchmaker Rye IPA. The hoppy flavors from the beer paired with the hoppy flavors from aioli didn't enhance each other, rather they don't each other down. In turn you could taste more of the rustic rye and malt character in the beer.
2. How to cut the fat.
For me anyways, eating a fatty and rich dish can be a bit tiresome after a while. Fat has a way of accumulating on the palate and needs to be cleared. This can be accomplished a few ways. First hop bitterness actually works similar to acidity in wine. The astringency of a really bitter IPA can really help clear fat off of your palate. This can also be accomplished with acidity from a sour beer, or a beer with high alcohol. The following is a picture of duck confit (actually cooked in pork and bacon fat) over a cannellini bean and olive oil puree with sage. That was paired with the Jester King/Mikkeller Beer Geek Rodeo. If you're not familiar with it it's the Mikkeller Beer Geek Breakfast (Imperial Stout) with chipotle chiles. The high alcohol really cut through the richness of the duck, plus the smokiness in the beer from the chipotle and the smokiness from the bacon toned each other down... which then increased the chocolate and roast characters of the beer.
3. Ease the heat.
I don't have a picture handy to illustrate this next one, but like a lot of richness... a lot of spice can get exhausting. Let's say you have some spicy kung-pao shrimp and it's getting to be too much the last thing you want to reach for is a bitter/hoppy beer like an IPA or pale ale. The hop bitterness will actually magnify the heat. Instead reach for a sweeter beer, particularly your sweeter fruit beers work well in this regard like Magic Hat #9. For darker, richer meats that are spicy I recommend reaching for something like a milk stout.
For the sake of not being exhausting I'll end it here, there's plenty more on pairing beer with food, but these few basics will make your next beer dinner a lot more fun!
Noble King Beer Aiolisome of Tom Mueller's info on olive oil.) Also please pay attention to the order in which the ingredients are added, if you don't do it just right your aioli will not emulsify.
4 cloves garlic
1 tbs. Dijon Mustard
2 egg yolks
1 cup Extra Virgin Olive Oil
1/2 cup Beer with good hop character but not too aggressive bitterness (Jester King Noble King Hoppy)
Juice of 1/2 lemon
Coarse Salt and Fresh Ground Pepper To Taste
In the work bowl of your food processor mince the garlic and add a small amount of salt to help break apart the garlic and soften it up. Add the egg yolks and dijon mustard and puree for 30 seconds. This helps to incorporate some air into the aioli which will aid in the consistency of the finished sauce. Add half of the olive oil very slowly with the food processor running. Add the lemon juice and 1/4 cup of the beer, pulse together. With the food processor running slowly add the olive oil and remaining beer, alternating between the two every few seconds. If the aioli is too thin just add more oil and keep the food processor running until it thickens up. Adjust seasoning with salt and pepper.