In a nutshell
For the travelling salesman problem — find the shortest tour that visits every city once and returns to the start — branch and bound grows partial tours city by city. At each partial tour it computes a lower bound: the least the tour could possibly cost to finish, namely the cost so far plus an optimistic estimate of the remaining edges. The moment that lower bound is no better than the best complete tour already found, the branch is pruned. The worst case is still exponential, but the bounds cut away most of the search.