Explicit elliptic fibration for $x_0^4+x_1^4+x_2^4+x_3^4=0$ (Fermat's quartic)

Question

In Huybrecht's Lectures on K3 surfaces, there's an explicit description of an elliptic fibration for the K3 surface $X:=\{(x_0:x_1:x_2:x_3)\in\Bbb{P}_\Bbb{C}^3\mid x_0^4+x_1^4+x_2^4+x_3^4=0\}$ (Fermat quartic), with the following explanation (example 3.11):

I don't get how proposition 3.10 helps us. My questions are:

1. How does the existence of the elliptic curve $E$ such that $mE\in|L|$ for some $m$ allow us to conclude that there is an elliptic fibration on $X$?

2. Was the map $X\to\Bbb{P}^1$ somehow suggested by the proposition?

3. How do I verify that the map is an elliptic fibration?