Math behind L2 Regularization for Logistic Regression
Your question is really about the method of Lagrange multipliers in constrained optimization, not logistic regression per se. The gist of it is that a constrained optimization problem can be recast as an unconstrained optimization problem by adding a term, called the regularizer, and vice versa. The sphere comes from recasting the unconstrained problem into a constrained one; recall that a constant $L_2$ norm defines a hypersphere.