Backtracking is a technique applied to reduce the search space of a problem. So, you have a problem, you have a space with optimal and non-optimal solutions, and you have to pick up one optimal solution.
A simple strategy, in your problem, is to generate all the possible solutions. However, this solution would traverse the entire space of solutions, and, some times, being aware that no optimal solution will be found.
That's the main role of
backtracking: you traverse the space of solutions and, when you reach a given point where you know no optimal answer will be achieved if the search continue on the same path, you can simply repent of the step taken, go back in the traversal, and select the step that comes right after the one you found to be helpless.
In your problem, since the nodes can be visited more than once, the idea is to maintain, for each vertex, a list of vertices sorted decreasingly by the distance from the vertex owner of the list.
Then, you can simply start in one of the vertices, and do the walk on the graph, vertex by vertex, always checking if the objective is still achievable, and backtracking in the solution whenever it's noticed that no solution will be possible from a certain point.