Although DNA is compact for the most part, they do unwind for replication and transcription. If Ethidium bromide comes in contact with such unwound DNA, they can slip in between the base pairs in the double helix and cause mutations.
But, according to this article, https://blogs.sciencemag.org/pipeline/archives/2016/04/18/the-myth-of-ethidium-bromide , EtBr is not a human/mice/rat mutagen. At least not at the concentrations we deal at the labs.
I think it just means that EtBr will understain densely compacted DNA. Likewise it also stains single stranded DNA poorly. This is probably because it is an intercollating agent, gets in between the DNA strands.