Latitudes and longitudes, like angles, are expressed in terms of degrees. Degrees are a way to measure angles. They tell us how much of a full rotation (a circle, in essence) the angle makes up. There are 360 degrees in a circle.
Longitude and latitude are expressed in this way because the Earth is a sphere. Any line drawn around the Earth, then, is a circle. The longitude and latitude essentially tell us the angle between the center of the Earth and a given spot on the globe.
Imagine that we have a spot that is the exact center of the Earth. We then draw a line from it to the Equator. Now we draw a line from the center of the Earth to another point somewhere between the Equator and a pole (north or south). This gives us an angle. The latitude is expressed as the measure (in degrees) of this angle. The same thing can be done for longitude, though in this case, we would draw a line from the center of the Earth to the Prime Meridian (arbitrarily placed in England) and then another from the center of the Earth to some other spot east or west of the Prime Meridian. This would give us our angle.
Thus, latitudes and longitudes are expressed in terms of degrees because degrees are parts of circles and lines running around the Earth are circles.