How does the two phase method for linear programs work... I understand that by adding artificial variables the problem can be reformulated as a new problem where the "starting point" is readily found.
What I don't get is how when this extended problem is minimised why it is a basic feasible solution to the original problem. 
 A: It isn't.  The objective of the new problem is constructed so that
1) Any feasible solution of the new problem has objective value $\ge 0$.
2) Feasible solutions of the new problem where the objective value is $0$ have all artificial variables $0$ and correspond to feasible solutions of the original problem. 
When you solve the new problem, it may be that the objective value of the optimal solution is not $0$, in which case you declare the original problem infeasible.  Otherwise, the objective value is $0$, and thus you have a feasible solution of the original.  
There is one technicality: it may be that you get a feasible solution that is not basic.  This would mean that one or more artificial variables is still in the basis though its value is $0$.   If the row of the simplex tableau for such a basic artificial variable
contains a nonzero entry for a non-artificial variable, do a pivot where that non-artificial variable enters the basis and the artificial variable leaves.  Because the value for that artificial variable was $0$, this is a degenerate pivot which won't ruin feasibility.  It is possible that this process will remove all artificial variables from the basis and leave you with a basis consisting of non-artificial variables, and thus a basic feasible solution.
However, it is also possible that you may get an artificial basic variable that can't be removed by this method, since all non-artificial variables have coefficient $0$ in its row.  What
that means is that there was a redundancy in the constraints.  You can just ignore this basic artificial variable and its row of the tableau; as long as all the other artificial variables have value $0$ this one will too.
