the hydrogens on the side chain substituent (the part branching out of the cyclic ring) are all on the same plane, right
no. the carbon atoms are still tetrahedral and sp
3 hybridized, thus the protons are still oriented in 3-dimensions. There is free rotation about both of the C-C bonds in the side chain. So if you arbitrarily position the two carbon atoms in the plane, then the two protons on each carbon atom are either in or out of the plane.
There are two reasons they coalesce into one signal. First, there is free rotation about the C-C bonds. Thus, the protons are not locked into one chemical environment the way the axial/equatorial protons are in the cyclohexane ring are.
Second, even though there is free rotation, if the protons are diastereotopic, they will give slightly different chemical signals. The classical way for determining if a set of protons are diastereotopic or not is to exchange one proton for a deuteron (a deuterium atom). If that forms a new stereocenter, then the two protons are stereotopic. If you exchange the other proton for a deuteron, you will either form the enantiomer of the the first deuterium-substituted analog, or a diastereomer of the first analog. If you make enantiomers, the protons are enantiotopic and give the same NMR signal. If they are diastereomers, they protons are diastereotopic and give different NMR signals.
In this case, the side chain protons on the first carbon atom can be substituted with deuterium to give enantiomers (the carbon center on the ring bearing the side chain is still not a stereocenter, so exchange with deuterium on the side chain only gives one stereocenter). Because exchange gives a pair of enantiomers, those protons are enantiotopic and will only give one signal.
If you exchange protons on the same carbon within the ring, you generate a stereocenter at the carbon bearing the protons being exchanged for deuterium... AND the ring carbon bearing the leaving group is now a stereocenter. There are now two stereocenters in the molecule where one of the ring protons is substituted for deuterium. Exchaning the other proton on that center gives a diastereomer of the first compound, thus the protons are diastereotopic and will give different signals in the NMR.