Theoretical comparison of dipole moments of N,N-dimethyl-4-nitroaniline and N,N,2,3,5,6-hexamethyl-4-nitroaniline
The workbook's explanation could have been more specific. If you build a 3D model, you may see that the orbital interaction between the lone pair on the $\ce{NMe_2}$ and the $\pi$-system on the ring is somewhat hindered by bad overlap because the steric interaction of the methyl groups (on the ring and the $\ce{N}$) prevents perfect alignment. Same for the nitro substituent. A higher interaction would lead to more charge shift.