Explain 0-extension algorithm
Page 1 explains how you can go from the $\delta$-based formulation to the $f$-based formulation and back. But it doesn't matter for the algorithm. The first step is to solve the LP, thus obtaining values for the $\delta(i, j)$ values. From a pseudocode standpoint, you solve this by calling your favorite LP solver.
$c$ is the cost per edge, and is part of the input.
Terminals are the elements of the tree T (also given as part of the input)
$A_u$ is defined in the line above the text 'Rounding procedure' as the minimum distance between u and elements of T. 
\sigma_j denotes the jth element of T in some random ordering of the elements (basically scramble the array of indices corresponding to T and take the jth element)