It could be any application where accurate quantification as well as purity of DNA sample is of importance. Could be cloning, chip-seq, PCR, qPCR or NGS. Since the protocol for these request for as pure DNA as possible.
Getting rid of RNA is especially important if you want to anneal the DNA to something. The problem is that the RNA may be complementary to the region of interest, and will then block your probe/primer from binding to the DNA. It is usually not a big concern if you plan to do a PCR because the primer is present in excess and has multiple possibilities (one per cycle) to bind the template. However if you try to probe your DNA or do binding assays with proteins contaminating RNA can interfere with your experiment.
Basically they need RNA free DNA because during concentration determination nanodrop gives combined concetration(DNA,RNA and protein) rather only DNA.Therefore, for downstream experiments one can get higher concentration of desire samples if it is not purified properly and hamperd the experiments specially if it is any sequencing.