Because you’re making all the incoming light rays pass through a tiny opening, which brings them into focus. It’s the same principle behind a pinhole camera. In theory, the smaller hole you make the more the image will be in focus, though in practice a smaller hole also limits the amount of light that passes through, which also makes the image dimmer.

Objects being out of focus is due to them being viewed from multiple locations, i.e., by all the points on the surface of the lens. Pinhole cameras, which have just one tiny hole instead of a lens, see everything as being in focus but, of course, the image is extremely dim. You need a lens to gather more light but then you need to choose the distance to focus on. The bigger the lens, the smaller the range of distances that are in focus at once. Photographers call this range of distances the "depth of field" and they know that adjusting the "aperture" of their lens (the size of the hole in the lens) will control the depth of field.

This is also why focusing is easier for people in bright sunlight, because then their pupils shrink down to a small hole. In dim light with large pupils, focusing needs to be more accurate.

As I understand it, it's because you are restricting the variance in the angles of light rays that enter your eye and hit the retina.

The way your eyes (and cameras) work is by using three different components to focus light to create an image. First, you have the retina, in the case of a camera a sensor, which is what captures the photons and creates the image. If you have just the sensor, then light hits it from every direction, and you just get a blur with varying intensity. So you have to do two things to get a clear image: you have to focus the light rays and you have to eliminate light rays that are interfering with the image. Focussing the light rays works by using a lens, which both your eye and a camera have. They refract the light rays in a way, so that they converge ideally at exactly the point where the sensor/retina is located, producing a sharp image. But this still leaves you with a problem. If you have a sensor and a lens, you can focus the light rays, but you also receive all of the light that is not being focussed. So the other thing that both a camera and your eye have is a small opening in front of the sensor, that restricts the angle at which light can enter and hit the sensor, in the case of a camera that is the aperture and in case of your eye that is the iris. If you have ever used a camera and are slightly familiar with the settings, you may know that decreasing the aperture (increasing the fstop) increases the depth of the area that is in focus. This happens because by making the hole smaller, you are cutting out some of the light rays, which means cutting out some of the variance in the angles at which light hits the sensor, which makes more of the image in focus (although this isn't the only factor that determines sharpness of course). And that is also what you are doing with your hand in front of your eye, decreasing the aperture, reducing the variance in the angles of light rays that enter your eye, therefore making up for what your lens may not be able to focus properly. This is actually how a camera obscura works. It uses no lens, but only a tiny opening relative to the "sensor", therefore cutting out all light except for the rays that enter at a very specific angle, and thus producing a relatively sharp image, if the sensor is the right distance from the opening.

This works because you're "pin hole" is filtering out "noise" by limiting the light coming into your eye to the light coming off of what you're aiming at only. Not because it focuses anything, your eye is still doing the focusing, it's just not having to deal with a bunch of extra light that isn't part of what you're looking at.

Things come into focus when the rays of light meet up at the same place at the back of your eye. Since all of these rays come from different directions, you need the lens in your eye to bend them to meet properly (or glasses/contacts that do the same).

If you look through a small hole directly in front of your eye, you're eliminating all of the rays of light except those that go straight into your eye and hit the back of your eye at the same point.

The eye is meant to image a point of light, for example a star, as precisely as possible on your retina so that you perceive it as, well, a point. And objects, like the text you try to read, will appear sharp if each of their individual points is imaged as a point.

Looking at a bright star without your spectacles you get an idea of how your eyes fail at that task: what should be a point is smeared out in various ways (depending on what opticians call aberrations of the optical system, or how well your eyes can focus, or both). This means that different rays of light (from the same star) entering different parts of an eye are sent not to the expected focus point but slightly nearby, and together adding to a blurry image of what should be a point.

Now by looking through a tiny gap you cut out most of these rays, which makes the blurry patch smaller, more like a point, and improves the sharpness of your vision.

Of course there are limits to that "stopping down": the image gets dimmer and, with a really small gap, the wavy nature of light actually starts increasing the blur again.

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A lens takes all the light rays coming from a certain direction and bends them so they all end up in the same place. In the case of your eye, they need to end up on the retina. When your pupil is larger, you have to bend the rays coming in at the edge more than the rays going through the center of your pupil. Terrible vision (usually) happens because the lens of your eye isn't bending the light rays by the right amount. Let's say your lens only bends them half as much as it should. Light going through the center of your pupil isn't affected because it they go straight through. Rays near the center get bent by the wrong amount, but because they weren't getting bent very much still end up in nearly where they should have. But rays that come through the edge of your pupil need to get bent by a lot, so when your lens isn't working right, they end up a long ways from where they should. That makes your vision fuzzy, because the light from say a single light bulb ends up spread all over your retina. If you block the outer parts of your pupil, though, those rays that ended up a longs ways off and made your vision fuzzy get blocked, so the light that does make it through ends up where it is supposed to.

The limiting case of this is, as people have mentioned, a pinhole camera. The pinhole doesn't bend the light at all, so what you end up with is a picture that has been smeared out by the size of the pinhole. The smaller the pinhole, the sharper (and fainter) the image. The eye is better than a pinhole because it's at least trying to focus, but it is the same basic idea.

Sort of. A true pinhole camera creates an image that is already 100% focused. The bigger the hole (“aperture”), the wider range of angles the light is coming in at, and the blurrier it gets without additional focus. (It’s not “noise,” it’s the wrong signal.) Think of holding up a piece of paper, take a little square on it, and imagine where the light comes from that hits that square. If there is no pinhole at all, and it’s just open to the environment, the light that hits that square is coming from all directions and you just see white. If you block out all but a medium sized hole, the light can only come from the direction of the hole, but there are still several possibilities for its source which get mixed together in a blur. Once you shrink down to an infinitely tiny hole, the light hitting any spot on the paper can only come exactly from the direction of one point on the other side of the hole. Repeat for every other point on the paper, and it’s in focus and you get a picture.

The problem is that an infinitely small hole lets in no light. And the wider the hole, the more possible angles that light can come in at, and the blurrier it gets. That’s where lenses or refractive mirrors come in, but OP didn’t ask about that…

So I didn’t know this was a thing, it kind of made me realize how bad my vision has gotten seeing the contrast

Just to add this good explanation...

So if you focus on a specific point, objects closer than that point will blur as they get closer, and similarly objects further away will blur as distance increases.

The range of distances over which you have acceptable focus is known as 'Depth of Field' (or DoF) and it's affected by two factors: 1.> How far away you are focusing (the further away you focus, the larger the DoF) and 2.> The size of the aperture (e.g. iris or a gap in your fingers) through which you're looking (the smaller the aperture, the larger the DoF - this is for the reasons explained by u/waitforthestopsign)

The DoF is not symmetrical - in fact objects closer to you than the focal point quickly blur (but we tend not to notice because these objects are often in our peripheral vision); objects further away than the focal point blur more slowly as distance increases.

A consequence of this is that in very bright light the iris in our eyes contracts to limit the amount of light entering the eye, which has the effect of increasing the DoF and makes it much easier for the lens to focus - this is because it the lens doesn't need to be as accurate, and deficiencies in the lens (e.g. long-sightedness) become less noticeable. The effect is particularly noticeable on objects that are close to us which is one reason why reading in good light can be so much easier that in poor light.

Bingo. This is why a pinhole camera can focus light sources at any distance.

Tradeoff: they are dim.