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By definition, a prime lens is a fixed lens system with a fixed focal length.
Then, simple physics tells us that it should be able to focus only on one plane (at a fixed distance) in front of it. But in fact you can focus on objects near as well as far.
What am I missing here?
A prime lens still has a moving focus element allowing you to change the focal plane through the range of the focusing ring's range. A prime is a lens that has a fixed focal length (100mm, 50mm etc) as opposed to a zoom which will allow you to cover a range of focal lengths (70mm-200mm for example).
A fixed focus lens cannot change its focal plane, but this is not the same as a prime lens.
This is a fundamental lack of understanding of what focal length means. Start with a simple, single-element lens. Hold it far enough away from the plane for something at "infinite" distance (say, the sun or moon) to be in focus. The distance from the lens to whatever you're focusing on is the focal length of that lens.
The focal length, however, is a direct consequence of how much that lens refracts light, which (at least mostly) depends on two things: the index of refraction of that glass/plastic/whatever in the lens, and the curvature of the lens surfaces.
To change the focus of the lens, you move the lens relative to the plane on which you're forming the image. Specifically, to focus on an object that's closer, you move the lens so it's farther way from the focus plane.
Given the normal situation where the lens will project an image slightly larger than the sensor/film plane, as you move the lens farther way from the focal plane, the angle of view recorded on the focal plane will shrink. In reality this is not a change in the angle of view provided by the lens, merely a reduction in the part of the angle of view provided by the lens that your sensor/film is able to record:
Here, the grey lines represent the picture being projected by the lens. The lower one represents what we'd get with it focused at infinity, and the upper with it focused considerably closer. The red lines in the upper show the narrower apparent angle of view due to the closer focusing. Note, however, that this is really a matter of not capturing the whole picture being projected by the lens, not a matter of the lens itself having a narrower angle of view.
For what it's worth, the same effect accounts for the effective aperture getting smaller as you focus closer -- the light that's projected past the edges of the sensor obviously isn't projected onto the sensor, so the closer you focus, the less light gets focused onto the sensor, so the smaller amount of light from the center portion of the picture is spread over the full area of the center. This means there's less light at any one part of the sensor, thus the smaller effective aperture (e.g., with a typical macro lens at 1:1, you lose almost 2 full stops, so a lens rated at f/2.8 needs roughly an f/5.6 exposure).
There's another effect to keep in mind as well: a lens that does internal focusing is basically also a zoom -- i.e., it does change the focal length (and corresponding angle of view through the lens) as you focus. A few (e.g., the current 105 Micro-Nikkor) are designed to have these to effects counteract each other (mostly anyway) so you maintain roughly the same real angle of view, regardless of where you're focused.
First, focal length is a property of a lens (by lens I mean a piece of plastic or glass that's inside your camera's photographic lens system). If you just have a single lens (think magnifying glass) and move it around in relation to an object and your projection plane (or a sensor in a camera for that matter), the focal length of that lens remains the same. The focal length is decided during the manufacturing of the lens (this is not the case with human eye, which can change focal length using muscles).
However, with the same lens you can focus on objects at different distances using this formula:
1/f = 1/s1 + 1/s2
Where f is the focal length of your lens, s1 is the distance from lens centre to the projection plane, and s2 is the distance from lens centre to an object. To focus at an object at a different distance you just need to adjust the position of the lens with respect to the sensor accordingly. Once again, focal length f always stays the same.
From this it follows that if you are taking a picture of an object that is infinitely far away 1/f = 1/s1 + 0 then your focal length matches the distance to the sensor, but for any other distance it will not be the case.
So I think the confusion arises from looking at prime lenses as a fixed system. Prime lens systems can change the distance from the actual lens to the sensor, but that doesn't change the property of that lens called focal length, which I could also define in this context as the distance from the centre of the lens to the sensor when an object that is infinitely far away is in focus.
I don't think the other answers really made it clear to me and did not resolve the confusion.
Then, simple physics tells us that it should be able to focus only on one plane (at a fixed distance) in front of it. But in fact you can focus on objects near as well as far... What am I missing here?
The part you're missing is that the focal length of a lens is measured when the lens is focused at infinity.
Focal length is the distance from the optical center of the lens to the image plane when the lens is focused at infinity. If you focus on something closer, then the distance from center to image plane may also change, but that doesn't mean that it's not a prime lens. The difference between a prime lens and a zoom is that the center to image plane distance is always the same for a prime lens when the lens is focused at infinity, whereas that distance for a zoom can be anywhere in a range of values when the lens is focused at infinity.
