2 cases:
If the graph has negative cycles: then the shortest path solution is undefined, because you can make a path of infinitely negative weight by reaching the negative cycle and traversing it an infinite number of times.
If the graph does not have negative cycles: then every cycle has non-negative weight. Then if any shortest path solution exists, there must exist some shortest path solution that does not contain a cycle. Suppose a shortest path solution S exists with a cycle C. We know that in this case C has non-negative weight. Hence we can remove C from S to get a path of lower or equal weight S'. Keep applying this process to S' until you get an acyclic path P. P has weight no greater than S, so it is a shortest path, and it is acyclic.