I suppose one of the best approaches (In my humble view, the best one) to delve into the concept of the degree of a projective variety is the method that has been used in "Algebraic Geometry" by Robin Hartshorne.
More precisely, if you study Chapter 1, Section 7 (Intersections in Projective Space), it will give you a real insight into the matter.
According to the fact that, as it is mentioned in the aforementioned part of the book, a purely algebraic definition of degree has been given, it would be much easier(at least to me) to gain a clear answer to your question.
I would like to mention that Theorem 7.5 ( Hilbert- Serre) plays a vital role in the aforesaid section and if you are already familiar with the Hilbert polynomial of a graded module over a polynomial ring and its Hilbert function, you can directly go to Hilbert- Serre Theorem and the rest of the section. Otherwise have a quick look at the whole section and in a few minutes you will get what you want.
I hope it will help.