Considering the definition that a telephoto lens is one whose length is shorter than its focal length:
Why do telephoto lenses (tend to) have long focal length?
Why not a telephoto lens with a short focal length (eg. 20mm or 10mm)?
Let's put it the other way round:
If you were to construct a lens for a desired focal length, under which circumstances would you use a telephoto lens construction?
To make it concrete, let's talk about the standard 35mm format ("full-size" 24mm x 36mm sensor).
For a 28mm focal length, why would you try to make the lens shorter than 28mm? I don't see a significant benefit, but that length constraint would make the construction more difficult.
But for a 300mm focal length, customers will highly appreciate a lens that is shorter than that, so it makes sense to use the "telephoto tricks" when constructing the optics.
Let us say we want a focal length of 10mm.
In order to be "shorter than its focal length", our lens design needs to be shorter than 10mm and still be large enough to illuminate our sensor. That is ... ambitious.
On a traditional SLR, the lens mount is about 40-50mm from the sensor. So we need to put the lens far inside the camera, directly in front of the sensor (<10mm). It will collide with the mirror. Also, the field of view will be obstructed by the lens mount.
To use a telephoto design for 10mm, you need a very small camera, e.g. the ones used in mobile phones.
Law of physics
In the image above, assume that the "object" is sufficiently far way from the camera so that the distance between the sensor plane where the "Image" is projected and the (ideally thin) lens is the focal length of the lens (or very close to).
The important thing to note is that the (red) light ray from the object that goes through the center of the lens is not deviated, so there is a straight line between the tip of the object, the center of the lens, and the tip of the image of on the sensor, so the two pink triangles are "similar", and the size of the projected image is proportional to the distance between lens and sensor.
Using a lens with a longer focal length lets you put the sensor further away to the right (or the lens further to the left), increasing the size of the projected image:
Ironically enough, a 10 mm or 20 mm lens (on a full frame or crop sensor DSLR body) sort of is a telephoto lens, in the sense that it has be a retrofocus lens (in order to not intrude in the mirror box). A retrofocus lens just uses a reverse telephoto group on the exit side of the lens, as opposed to a telephoto group on the front of telephoto lenses.
The primary benefit of a telephoto lens is that it is shorter than its focal length. But the lens still needs to be long enough to incorporate the necessary lenses to achieve its purposes.
Finally, note that wide angle lenses like 10 or 20 mm on a full-frame or crop body are already so short that their entrance pupil is very near, or even in front of the lens body. Especially for ultra-wide angle lenses, which by definition the entrance pupil must be visible from 90º to the side. Adding a telephoto group to an ultrawide angle lens pushes the entrance pupil rearwards, thus completely obscuring the wide angle of view the lens was trying to achieve. Telephoto groups (consisting of a strong convex element followed by a strong concave element) is mutually exclusive to how wide angle lenses achieve their angles of view: multiple concave elements up front, no exceptions).