The degree of a variety $X$ of dimension $r$ is defined by $r!$ times the leading coefficient of its Hilbert polynomial. This is the defination given in Hartshorne, but I find it is very hard to handle with. If $X$ is a curve, one can cut it by generic hyperplane, and count the number of intersection points. This gives the degree of curve. However, for the surface, I want to understand the following claim used by Hartshorne:
(1) $X$ is a surface with embedding $X \to \mathbb{P}^n$, and $D=O_X(1)$, then the degree of $X$ in $\mathbb{P}^n$ is $D^2$.
(2) $X$ is a surface with embedding $X \to \mathbb{P}^n$, and $D=O_X(1)$. Suppose $h$ is a divisor in $X$, then the degree of $h$ in $\mathbb{P}^n$(viewed as curves) is $D.h$
I certainly wish to see the rigorous proves, but presumably they will be long and not very enlightening. So any intuitive argument is also greatly welcome!