Your question is really "how does the human eye work?", since the contact lens is designed to adjust the optics of the human eye.
This image from the wikipedia article on the anatomical lens shows how the cornea and lens focuses incoming light from the left onto the retina (right).
Previously, I'd written that the biological lens in the eye was responsible for the optics; however, as pointed out in the comment of David White below, the cornea actually performs most of the focusing; according to wikipedia, it contributes about 2/3 of the focusing power. Most of the rest is due to the lens, which also performs an important role in the process of accomodation, which is how we change our focus between objects at different distances.
For people who are nearsighted or farsighted, the light coming into the eye does not end up in focus. See this picture for the case of nearsightedness:
You should imagine the contact lens being placed over the cornea (left surface) and causing the rays to adjust so that the image ends up in focus.
It's not quite clear what you mean by "telescope lens" - do you mean the system of lenses that make up a telescope? If so, there are two basic types. The actual lenses in your telescope are probably more complicated and correct for all kinds of aberrations, but they work like this.
The Keplerian telescope (top one in the diagram) consists of two positive lenses, with different focal distances, with their foci at the same point, in between the lenses. Imagine your eye on the left. Two parallel rays will converge to a point at the focus of the right-hand lens, and since this point is also at the focus of the left-hand lens, they will become parallel again, but inverted. Of course you are usually not looking at parallel rays with your telescope, but picturing it this way is what helped me to understand why we see a magnified image.
You might think at first glance from the diagram that this makes the image smaller; but what it does is take parallel rays traveling at different angles to the optical axis, and make them parallel again on the other side of the telescope, but traveling at a larger angle. This increases the apparent size of the object. (Google Docs isn't very good for drawing detailed diagrams - you could take a look at the more complicated ones on the Wikipedia page on telescopes.)
The Galilean telescope (bottom one in the diagram) consists of a negative and a positive lens, again with their foci at the same point. This time the point is on the outside of the telescope, at where your eye is. The positive lens focuses the parallel rays to that point, and the negative lens takes the converging rays and makes them parallel again. This time, the image is not inverted.
Then the second part of your question is about magnification. The focal lengths of the lenses are the only factors that influence the magnification: it is equal to
$$M=-\frac{f_2}{f_1}$$
The minus sign seems counter-intuitive, but think about it - we fill in a negative focal length for the negative lens in the Galilean telescope, so the magnification comes out positive. For the Keplerian telescope, the magnification comes out negative - this indicates that the image is magnified, but also inverted.
Best Answer
the autorefractor projects an image into the eye. the light rays pass through the lens and strike the retina. a small amount of the light bounces off the retina, passes through the lens a second time and exits the eye. Imperfections in the shape of the eye's lens distort and defocus the "return" image. the autorefractor senses the distortions and misfocus and tweaks the projected image with lenses of its own until the "return" image is in focus and distortion-free. the software in the autorefractor then deconvolves the tweaks and back-calculates the corrective lens prescription that the lens in the eye needs to properly focus the image.