The 4-cube has 16 vertices and 32 edges (4 edges meet at each vertex, and each edge connects 2 vertices). If the graph is planar then Euler's formula v–e+f=2 tells you that there must be 18 faces. But each face of the cube has 4 edges, and each edge is shared between 2 faces, so 18 faces would need 36 edges—contradiction, so the graph is not planar.

If n>4 then the graph of the n-cube contains the graph of the 4-cube as a subgraph. So the n-cube's graph must also be non-planar.