Bokmål actually means book language. When Denmark and Norway one one country danish was the administrative language and the one used in law texts and spoken in towns and cities.
We are talking about once upon a time when danish was still an understandable language
This language is what turned into standard norwegian/bokmål and it is very similar to danish, especially when written.
Nynorsk means new norwegian but isn't really new, it is just the norwegian spoken outside the towns. When Norway became a country of its own people started using it more to signal that they were not danish, but actually had a language of their own. Bokmål is closer to danish than nynorsk is, and thus easier to understand for danes at least.
So what is the story with you and swedish?