STAR-CCM+ stores the mesh connectivity as an array of cells adjacent to each face along with a list of vertices belonging to the face (in a valid mesh each cell interior face belongs to exactly two cells). Invalid cell and vertex references arise when the arrays storing the cell connectivity and vertices have entries that make reference to additional faces or vertices respectively that fall outside the expected range of values for the cell.
As for unclosed cells, an invalid cell or vertex reference are usually caused by the mesh exporter/importer or during mesh generation and have to be fixed before proceeding any further. Deleting the cell and replacing it is usually the quickest and easiest solution to this problem.