The most common answers to this question are:
1. They gather light, make dimmer objects brighter and visible to the naked eye.
2. They magnify and thus make small objects visible.
The first answer is mostly wrong (but not quite), the second however, surprisingly accurate though probably in a way that is unknown to most people. Moreover, a lot of people know how to calculate the magnification, but don't know how the telescope achieves it. I will try to explain these things below.
Colors will be ignored as they complicate matters and are not necessary for the basic understanding.
Before we can attempt to answer the first question, there is a question we should answer first: "How do we see things?".
It takes two properties to see things: there must be contrast and there must be size.
Imagine a situation where everything is a uniform gray (or other color). Since you only see gray in every direction you cannot actually see anything. It could be argued that you see gray, but if it is the only thing you ever see, you would have no way of knowing that you were actually seeing it.
Now suppose that in one direction the shade of gray is lighter. Would you be able to tell the difference (i.e. "see") it ?. Not necessarily. If the intensity difference is minimal and the two area's blend into each other gradually, then you cannot tell the difference. If the border between the two area's is sharp, then it becomes much easier to tell them apart. The eye/brain is excellent at detecting lines. As the borderline gets more and more diffuse, a higher contrast is needed to see the difference in intensity.
The second property (size) looks much easier to understand. When an object is to small it cannot be seen. However, contrast also plays its game here. An almost infinitely small spot (star) can be seen if it is bright enough. But as we all know, dimmer stars are harder to see. Also, the limiting magnitude of the stars you can see is not the same on every night. If the seeing is bad and/or the there is a lot of moisture in the air then the light of a star gets smeared out over a small area. The bigger the area, the less bright it will be (because the amount of light from the star is constant). This example illustrates that the combination of brightness and size determines the detectability (i.e. if we can see it). In other words: we can see objects that are small and bright or dim and large. The dimmer they are, the larger they must be. (Note: Mel Bartels has a nice piece on this, called "Optimum Detection Magnification", see "Links")
How do we perceive 'size' ?: Size is no more than seeing light coming from the sides of an extended object in two directions. The angle between these two 'directions' is the angular size of the object. In astronomy we often talk about angular size to differentiate it from actual size. To determine the actual size, you have to know the distance as well. The angular size is the ONLY size we can see, even for earthbound everyday objects. But for our everyday objects, the brain immediately fills in the actual size because it has computed the distance or has something to compare it with.
The Night Sky
When we look up into the night sky, we see stars (etc.) in a black sky. But just how black is "Black" ?. Even though we don't see anything, there is still a background brightness. It has been found that in truly dark places this can be of magnitude 21 or in special cases (high altitude) even better. If you have been in a (big) city, you probably know that it can be magnitude 3 or worse there.
How does a telescope work ?
There is one key word to this question: Magnification.
Telescopes are used for two different kind of objects; pinpoint objects (stars) and extended objects.
Pinpoint objects: For these objects magnification is used to 'thin' out the background brightness, while it leaves the brightness of the pinpoint object concentrated in that pinpoint. The result is a greater contrast between the object and the background, resulting in a better detectability.The background light can be considered as a certain amount of light per area. If by magnification the area appears bigger, the light gets more spread out so the surface brightness will be lower. A pinpoint object will stay a pinpoint object, even when magnified. Thus the contrast between the object and the background gets better with increased magnification.
Note 1: So why do bigger scopes show fainter stars ?. Well, because the usable magnification range depends on the aperture of the scope. The bigger the scope, the higher the minimum and maximum magnification. BTW, you can easily check this effect. Put a high power eyepiece in your scope and take a look. Remember the place of the faintest star you see. No put in a low power eyepiece and check if you can still see that star. (Remember this is still the same scope, the aperture did not change, only the magnification. So the visibility of that faint star depended on magnification and NOT aperture)
Note 2: Unfortunately the magnification can not be increased forever. When it reaches 50 times the aperture in inches (or 2 times the aperture in mm) the pinpoint nature of a pinpoint object disappears. It becomes a small disk with so-called 'diffraction' rings. This disk is now increased with still higher magnifications, so now its surface brightness decreases. See extended objects (below) for more on this.
Extended objects: The most surprising property is that extended objects do not get brighter in a scope. Also the contrast does not change (actually it only can get worse). The only thing that changes is that the object itself gets bigger by magnification. The eye/brain finds it easier to detect large faint objects, so the object itself gets better visible.When we look up at the night sky with a fully opened pupil, the pupil is about 6-7 mm in diameter. If we now use a scope with twice the opening (i.e. 14 mm) then this light has to be compressed by the telescope into the same pupil diameter (called exit pupil). But the difference in aperture and exit pupil is also the magnification of a scope. Thus twice the aperture of the pupil yields a magnification of two. But a magnification of two also reduces the surface brightness by the very same factor. The result is that no matter how large the aperture of the scope, as soon as we compress the collected light it in an exit pupil of 6-7 mm diameter we have the same surface brightness as without the scope.
Note 1: If the light is compressed in a smaller exit pupil (as is often the case) then the surface brightness actually decreases, but the contrast remains the same.
How does a telescope magnify ?
Since magnification is such an important aspect, lets take a short look at how it is achieved. Below the light path of a scope is shown for an on-axis pinpoint object. It is shown for a refractor, since the drawing for a reflector would be more complex. The principle is the same.
Unfortunately the drawing is not to scale so don't use it for designing you 1.5" scope :-)
In the drawing above the light enters the objective (from the right) and is brought to a focus at the distance Fo (the focal length of the objective). The light continues its path until it strikes the objective after a distance Fe (the focal length of the eyepiece). After the eyepiece the light again forms a parallel bundle, the diameter of which is called the exit pupil.
Magnification is show in the figure below:
The red lines illustrate a light bundle coming from the 'top' of an extended object. The green lines from the 'bottom'. The same thing happens as in the figure for a pinpoint object, but now the red and green light bundle exits under an angle from the eyepiece. The angle between the incoming red and green light bundle and the angle of the outgoing red and green light bundle is different. In fact this difference is the magnification.
In order to see the whole image, the eye must be placed at the spot where the outgoing red and green light bundles cross each other. The distance between this point and the eyepiece is called eye-relief.