Space

Yes, but it's not just brighter near the outer edges, it's whiter, thanks to the inclusion of more blue light, and that less ruddy colour does a lot to affect how we see the gradation from light to dark.

They are harder to find than they should be, but there are pictures of the Earth occulting hte sun out there, and you can see how the Earth's atmosphere is aglow with scattered sunlight. There's more light closer to where the sun is than where isn't. The Blue Ghost lander pictures of the eclipse are probably the most accessible right now:

The bright ring around the Earth in ths image is sunlight streaming through the Earth's atmosphere. See how blue it looks? And how it gets brighter closer to the sun?

That's why.

Have you seen a sunset when you had a pretty clear view of the horizon such as over water or in unforested plains? Once the sun is fully hidden from view, there is still the sunset gradient in the sky. Red on the horizon, orange above, yellow above, and, for truly most of the sky, a pale blue. Without clouds to give dramatic clashes of alternate colors^1 , the sunset color band is very slim, as evidenced by trying to use a phone camera to take a picture.

Now think about where the sun, earth, and moon are in space. You're probably thinking of it wrong. The sun is extraordinarily large but also extraordinarily far. So far, in fact, that it appears the same size regardless of whether you're on the earth, on the moon during a solar eclipse, or on the moon during a lunar eclipse. The variation is less than what we experience during a supermoon. While the Earth is much larger than the moon, they are significantly far apart for their size. Think of the picture of Earth taken by the Apollo astronauts. It's pretty small, right? The Earth is about 4x bigger than the moon in diameter. Since, from Earth, the moon and sun both appear to be the same size (0.5deg of view across), we can estimate the Earth appears to be 2 degrees wide as seen from the moon. It can completely block the sun.

But the Earth has a trick up its sleeve: the atmosphere. It refracts light. While we sit on a beach and watch the sunset over the ocean horizon, that color band is fairly narrow in horizontal width and incredibly short in vertical height. It feels like a pretty small event if you use a wide angle camera. But it's not. The red band encompasses the Earth all the way around. When there's a sunset seen from California, there's a sunrise seen from India. Say we're at an equinox, which means both North and South poles are also in a golden twilight. The sun lights half the planet at any given time, meaning that red/orange band of sunset colors is actually an entire ring around the planet all the time. Not all of that light lands on a photographer's sandy toes. In fact, most of it passes all landmass and continues into space. With the blue light scattering in more directions that red, much of it gets reflected away from Earth and into space at all angles. Meanwhile, the red end of the spectrum fares better and continues straighter - but it DOES scatter to some extent, including into the shadow of the Earth. In the grand scheme of things, this is insignificant and unnoticeable. But, for us, from our unique vantage point between 3 bodies of shockingly similar apparent (visual) size, the effect is that the ring of red sunset/sunrise glow blurs the Earth s shadow enough to create a refracted red "shadow" onto the moon. But, as mentioned above, that halo of red refraction is still slim and quickly gives way to the oranges, yellows, and pale blues. That blends on the moon somewhat, making the red portion lean to one side while the opposing side appears more balanced in color.

^1 clouds are what make subsets look dramatic with the clashes of pinks and oranges and such. It's not actually pollution. Pollution only works to dull out sunsets. Since clouds are at such great altitudes, they "experience" sunset a little bit later than a ground observer. This means while you see a red sun, the clouds see an orange sun. When you see orange sky, the clouds see yellow. The difference could be greater if you have higher clouds or closer if they're lower. Or, of you're lucky, all different altitudes such as when you see red sunset, fluffy orange clouds, and streaky yellow clouds in the Jetstream, all set in front of the pale blue sky. Clouds may block the view of the actual sunset, but they're responsible for the most breathtaking sunset views.

Go out on a sunny day. Get in the shadow of a building - near the edge. Notice that you can know when the sun is close to becoming visible by the scattered light. One side is brighter.

Now imagine being in orbit of earth. Watch the sun set behind it. You see the sky glowing in the direction of the sun. Same as the building. Same as regular sunsets.

Now go to the moon. Because of the extreme distance, you can see the "sunset" entirely around the globe. That is the red glow. But one side of the moon is closer to the sunset, and (just like the building) that side is brighter.

Does that help?