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I understand how a prism works and how a single raindrop can scatter white light into a rainbow, but it seems to me that in normal atmospheric conditions, we should not be able to see rainbows.

When multiple raindrops are side-by-side, their emitted spectra will overlap. An observer at X will see light re-mixed from various originating raindrops. The volume of rain producing a rainbow typically has an angular diameter at least as wide as the rainbow itself, does it not?

So why can we still see separate colours?

Your picture shows that an observer at X will see both the red and the blue scattered light, but he will see it coming from different directions.

That's the same way you usually see things: different amounts or colors of light reaching your eyes from different directions, and thereby creating an image on your retina.

This isn't quite how rainbows work. The standard explanation is that light bounces around inside each droplet, and getting reflected once, and exiting at an angle:

^{Image source}

However, the real picture is a little bit more complicated. When sunlight hits a water droplet, the rays will

1. refract when they come in,


2. (partially) reflect back when they hit the back of the droplet, and then


3. (partially) refract on their way out.

For each droplet, though, there are a bunch of rays hitting the droplet at different locations, and each of them will bounce around differently and exit at a different angle, so that the end result looks like this:

Because there is a reflection inside the droplet, the light is mostly sent backwards, and because there are two steps where refraction happens, the angles are a bit wonky. But here's the important thing: the angle at which the light exits increases, has a maximum, and then decreases again, a fact which is clearly visible by following the dots as they go down from the negative-$x$ axis, stop, and then go back up again.

This means that if the relative angle between the Sun, the droplet, and your head is smaller than a certain maximal angle $theta_mathrmmax$, usually equal to about $theta_mathrmmaxapprox 42°$, then the droplet will appear bright to you (and, since this isn't an individual droplet but a misty conglomerate, the mist will have a diffuse glow), and if the angle is larger than that, then there will be no extra light going towards your eyes from those droplets.

In other words, then, this process will produce a disk that's bright, centered at the anti-solar point (i.e. where your eyes receive the on-axis reflections in the diagram above) and with diameter $theta_mathrmmaxapprox 42°$, and this is precisely what's observed, particularly when the rainbow happens against a darker background:

^{Image source}

Notice, in particular, that the inside of the (primary) rainbow is much brighter than the outside.

Moreover, notice that the brightness of this disk increases as you go from the center to the edge: this is caused because the rays cluster at the turning point at $theta_mathrmmax$ (notice in the ray diagram that there's many more dots in that region than there are near the axis). This clustering means that, for each color, the disk of light has a particularly bright edge, called a caustic.

So what's with the colors?

Although your diagram's geometry is off, as you correctly note, the standard diagram (the first figure in this answer) is kind of misleading, because for it kind of implies that for every red ray that hits your eyes, there will be another droplet at another angle sending a yellow ray (or green, blue, orange, indigo, and so on) on the same path ─ and that is indeed correct! This is what happens inside this disk of light.

The thing with this process, though, is that the maximal angle of aperture of the cone of light that's reflected by each droplet depends very sensitively on the refractive index of the water that makes up the droplet, and this refractive index also depends on the wavelength of the light, so that the size of the disk increases with the wavelength, with the red disk being the largest, then the orange, yellow, green, blue, indigo and violet being successively smaller.

This means that, at the edge of the disk produced by the red light, where it is the brightest, there is no light of other colours to compete with it, so the light looks red there.

A bit closer in, at the edge of the orange disk, there is no light of yellow, green, or blue colors, since those disks are smaller ─ and, also, the light from the red disk is fainter, because it's not at the maximal-brightness edge and the orange disk does have its maximum shine there. Thus, at that location, the orange light wins out, and the light looks overall orange.

And so on down the line: for each color in the spectrum, the edge of the disk is brighter than the larger disks, and the smaller disks don't contribute at all, so the edge of each disk shines with its respective color.