Often when people talk about using light-meter, they mention that it has the ability of measuring light at X degree angle.
What does it mean?
How can we make use of that information to measure the light of our subject accurately?
Those talking about the 'angle' refer to the angle of view of light meters.
A standard reflective light meter has a fairly large angle of view, and thus meters for a large area in your scene.
Spot meters have a very narrow angle of view. You've likely seen mentions of 1-degree spot meters. This means that such a spot meter meters for 1 degree in your entire scene, and is therefore a very precise piece of equipment (that is, it is precise for a very specific area):
You can see how the circle in the spot meter's viewfinder, which indicates the measuring area or angle of view, is rather small. For other (in-camera) reflective light meters, this area would be significantly larger, such as 15 degrees.
Note: this circle is not to be confused with the split-image circle many older film cameras have.
In addition to timvrhn's answer above (great answer, btw). You also asked "How do we make use of that information?"
When shooting a scene (e.g. a landscape) you may be dealing with a potential broad exposure range.
Having a narrow "spot" focus feature allows the photographer to more easily isolate the brightest and darkest features in a scene to determine the entire exposure range.
E.g. is the brightest element in the scene say... 10 stops brighter than the darkest spot in the scene? Does the camera easily deal with those 10 stops of range?
If yes then they can use the meter to determine the median exposure (the best exposure to capture the entire range without any clipping of information).
If no then they can use that information to determine the range of exposures they should capture should they want to use bracketing to create a high-dynamic-range image.
Very often landscape photography can push the limits of a camera sensor's dynamic range by require more exposure range than the camera can natively provide.
Suppose the brightest element in a landscape scene is the top of a white puffy cloud. And suppose the darkest point is a shadowy area in a deep ravine. You could meter those two points to find the full exposure range necessary for the shot. But you need to be able to isolate just the very brightest point and just the very darkest point.
But now suppose your meter can only offer a 3° ... or maybe a 5° spot. Instead of isolating just the very brightest point or very darkest point... you're getting an average of all the light in the area that fits within the spot. This means the brightest point and darkest point probably require an exposure a little beyond the meter reading. A smaller target (narrower angle spot) makes it easier to isolate these points.