Hi, lonelygoatherd--
The trick is to remember that diastereotopic hydrogens show up differently on NMR whereas homotopic and enantiotopic hydrogens show up the same on NMR. Once that chloride was added, you will be making diastereomers if you replaced any of the non-CH3 hydrogens with a different atom.
For example, if we assume the chloride is going back (it looks like the H on the carbon sharing the chlorine is wedged), that carbon is R configuration. Now, move to the CH2 to the right of that carbon containing the chlorine. If you replaced one of those hydrogens with a chlorine, you could get R,R dichloro. However, replacing the other hydrogen would give R,S dichloro. RR and RS are diastereomers, so the two hydrogens show up differently on NMR. If the chlorine was not on the original compound, there would be 6 peaks (because you would only get R or S if you exchanged a hydrogen for a chlorine, and R vs. S are enantiomers and not diastereomers). This picture may help:
View attachment 234994 One is RR and the other is RS (or SR and SS...depending on how you read the original one). Swapping a terminal H (from the CH3) will not cause diastereomers because a CClH2 is not stereogenic/chiral.
Does that help?
Happy studying!