Picking mesh elements that are not on the border of the mesh

vm = VoronoiMesh[pts, {-1, 1}]
HighlightMesh[vm, MeshCellIndex[vm, {2, "Interior"}]]

Related: Boundary cells of a mesh?

Show[vm, Epilog -> {Opacity[.7, Orange], MeshPrimitives[vm, {2, "Interior"}]}]

For planar MeshRegion that arise from DelaunayMesh or VoronoiMesh, usually

R["InteriorFaces"]

should work.

A more general and more transparent ways is to use the package "IGraphM`" by Szabolcs as follows:

Needs["IGraphM`"]

A = IGMeshCellAdjacencyMatrix[R, 1, 2];
bndedges = Random`Private`PositionsOf[Total[A, {2}], 1];
interiorfaces = Random`Private`PositionsOf[Total[A[[bndedges]]], 0];

HighlightMesh[R, Thread[{2, interiorfaces}]]