A "supermoon" is what the media likes to call a full moon when it's near the perigee of its orbit, i.e. it's at its closest approach to the Earth. The Moon gets this close to the Earth once during each orbit of the Moon around the Earth, so about once every 28 days; thus the Moon appearing this large in and of itself is nothing special, but I suppose it makes for feel-good fodder on slow news days. In actuality the Moon only appears slightly larger; about 14% larger than it appears at apogee (this is when it appears its smallest), or around 6-9% bigger than the average apparent size of the Moon. Because of this small difference, combined with the fact that we only see the moon intermittently over long periods of time, and in different phases, it is highly unlikely that the average person would naturally notice the Moon was larger or smaller from week to week.
There are a few reasons why people "notice" a supermoon being bigger than a regular full moon, and it all comes down to our imperfect human perception.
First, the news reminds us to look at the Moon because it will be bigger than usual. We go out and look, and because a bigger moon is what we're expecting, we agree "well, I guess it does sorta look bigger." This is a form of confirmation bias. In truth, most of us have no intuitive sense for how big the Moon should actually appear, certainly not to an accuracy that would make us notice that the moon was 5%, 10%, or even 14% larger than at some other time we saw it.
Secondly, a full moon rises during sunset because a full moon is always on the opposite side of the Earth from the sun. Most people are out and able to see the Moon around sunset during their evening commute, so when there's a full moon (or a supermoon), most people will see it as it's rising, close to the horizon. When the Moon is close to the horizon it always appears bigger regardless of where it is in its orbit -- this is an optical illusion of sorts because suddenly we see the Moon next to objects on land and its comparative size appears bigger. This problem, sometimes called the "moon illusion" is a long-known issue with human perception and has been discussed by scholars for thousands of years. Click here for more information on the "moon illusion." Typically, doctored photographs (of which there are many on social media after the media hypes a supermoon) fraudulently increase the size of the Moon almost to the point of ridiculousness, most likely because the photographer thought that the Moon "looked really big!" and is confused as to why their photographs don't reflect what the optical illusion had them perceive, regardless of the focal length the photo was captured at. The photograph they took is indeed accurate, but being a flat image, it just doesn't play into the same part of our brain that produces that optical illusion of the Moon looking huge next to the horizon. In actuality, any time you see the sun or the moon rise behind features on land, it will appear very big.
As a consequence of being marginally closer to the Earth, a supermoon also appears slightly brighter, owing to the fact that the flux (or density) of reflected photons from the surface of the Moon is higher when we get closer to it. An analogy for this would be that your shower feels more intense when you move your face closer to the shower head. However, I would argue that most people wouldn't notice this change in apparent brightness any more than they would notice an increase in apparent size. Our eyes involuntarily adjust to low light levels all the time, and it would be impossible to make an assertion that the Moon was brighter one night vs. another without using some additional equipment (a camera with manual exposure controls would suffice). Also, the clarity of the atmosphere (depending on temperature, humidity, particulates, etc) varies frequently, adding yet another variable into the situation that we humans are not well suited for evaluating without special equipment.
To me, supermoons are fun just because they get people interested in and talking about the Moon. The Moon, while not a favorite subject of mine in and of itself, is definitely one of my favorite compositional elements in a photograph, and all the overzealous reporting in the world won't change that.