I think this question needs a great deal of attention by the BW writers. Whatever they come up with, I do hope they will give us an extensive explanation for this. There are several ways, I could imagine how this issue could be addressed properly:
1. A gameplay prologue in the Milky Way: This may work, especially if the departure to Andromeda happens during the reaper war. An Intense fight and close escape could make for a cinematically impressive introduction of the new game while allowing us to do some more relaxed world building afterwards.
2. A novel or a CGI movie or some such (please no anime
). This would be the option that is most removed from the actual game. However, if BW does want to make the cut as clean as possible, I think it's a viable option. This way, the fans have the option to get the information while other players can just start out fresh without any baggage from the transition. It would also give the writers a (theoretically) infinite space to explain everything and allow for a very detailed explanation of a possibly complex scenario.
3. Integrate the explanation in the main plot or a side quest line: They could go with the good old cliche that our guys wake up from cryogenic sleep in Andromeda but no one really knows how exactly they got there. Finding out could be a quest line, which would provide enough room to explain how we got there in little snippets. Making it a mystery that needs to be solved can also add some tension to whatever the final explanation is going to be.
I'd be ok with any of these options but I do think we need some time to deal with the link between Andormeda and the milky way. A 30 second intro sequence and a codex entry, saying "you went through a worm hole, 'cause plot" will definitely not cut it for me.