The main goal of the Branch and Bound algorithm in Integer Linear Programming (ILP) is to find the optimal integer solution to an optimization problem by systematically exploring and eliminating portions of the solution space. It does this by dividing the problem into smaller subproblems (branching), calculating bounds on the best possible solutions within those subproblems, and pruning branches that cannot yield better solutions than the current best. This approach helps efficiently navigate the search space to identify the optimal integer solution while minimizing computational effort.