Why are we interested in negative edge weight cycles when finding out SSSP. In a path, we are not allowed to traverse a node more than once.
In a simple path, no node can be traversed more than once. However, paths in general can. The SSSP asks for single-source shortest paths, without specifying that the paths need to be simple. This allows the paths to contain cycles. Thus, negative cycles are a concern as they may allow arbitrarily "short" paths, and so the definition of a "shortest" path falls apart. This is because if a path contains a node on a negative cycle, we can always lower the cost of the path by going around the negative cycle.
Of course, this also depends on how you define a path, and how the SSSP has been presented to you. The above explanation assumes typical definitions of both, so if you have different definitions, please leave a comment.