People have been overstating the differences.
Between Danish and Norwegian, the written standards of the languages are very similar indeed. Reading Danish is like reading slightly old-fashioned Norwegian, with a few consistent misspellings, a tiny handful of unusual vocabulary, and too many commas. As for spoken Danish, nobody really understands that anymore, not even the Danes themselves. (Or rather, most dialects have drifted toward a rather slurred pronounciation, so it takes some extra familiarization to be able to decipher the spoken language. Danish kids are now, in fact, among the slowest in the world to master their own native language.)
Swedish is a bit more different both in terms of grammar and vocabulary, but only a bit. In return, the Swedes actually enunciate their words properly, so it is easy to hear what they're saying. For any native speaker of Norwegian (or Danish) it takes very little familiarization to get past most of the "silly misunderstandings" stage. Neighbouring (across the border) dialects of Norwegian and Swedish often resemble each other a lot, both in tone and in vocabulary (which gets swapped back and forth).