How do radians work? Why can't we just keep working with degrees in school?

Degrees are a made up unit; mathematicians simply decided that a complete revolution of a circle has 360 degrees, without that number actually meaning anything. Tecnically any number could have been picked to split a whole revolution into and, mathematically speaking, it would be just as valid as picking 360. On the other hand, radians split the circle into a number of segments that is not arbitrary: 2pi or around 6.283. Even though that number looks more complicated it makes more sense and it's actually related to what angles are. If you think about it, an angle is nothing more than how far you go around a circle. You can see this by looking at the length of the arc that you create when drawing a circle segment; the bigger the arc, the bigger your angle. You could draw the same segment of a circle with a bigger radius. The bigger you make the radius for that segment, the bigger you have to make the arc too, while the angle stays the same. If you look at these two quantities, arc length and radius, you will notice that the ratio between the two will always be the same if you keep the angle constant. So the ratio of arc length and radius after one half revolution actually gives you that fundamental number that shows up in everything that has to do with circles: pi. You might recognize that the equation to determine the perimeter of a circle is based on the same fundamental principle. One full revolution (perimeter) = 2pi * radius, or rewritten 2pi = perimeter/radius. Because the radian is intrinsically related to these basic geometric principles, it is much better to use in calculus and other areas of math as you move on.