Understanding StarCraft II without having played Starcraft I
Solution 1:
During the install, the Adjutant describes the events of the first game in an overview fashion to get you caught up. In addition, they do a good job of giving you key information from the first game throughout whenever they reference events from it.
Starcraft II does have its own plot that is not entirely dependent on the first game, so they went to lengths to ensure that you need not have played the first one to fully enjoy the second one.