We develop and analyze a new affine scaling Levenberg-Marquardt method with nonmonotonic interior backtracking line search technique for solving bound-constrained semismooth equations under local error bound conditions. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable. The global convergence results are developed in a very general setting of computing trial directions by a semismooth Levenberg-Marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set. We establish that close to the solution set the affine scaling interior Levenberg-Marquardt algorithm is shown to converge locally Q-superlinearly depending on the quality of the semismooth and Levenberg-Marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition, that is, BD-regular condition under nonsmooth case. A nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature.