The story of Romeo and Juliet, traditionally listed amongst the Tragedies of Shakespeare, more specifically can be considered a "disaster!" The phrase "star -cross'd lovers" of course refers to them both originating from warring households, that no matter what they felt for each other, the hard politics between their families would impede their love for each other, and cause the disaster of their suicides. "Disaster" is a Greek word, "Dis" meaning "against" and "Aster" meaning "star...."
To a Shakespearian audience, more versed in astrology and the effects the heavens had upon humans and their affairs than we are, the phrase "star-cross'd" would suggest to them a bit of foreshadowing of the trouble ahead.
This phrase refers to the idea that the stars determine people's destinies, like in astrology. So saying that they are "star cross'd" is saying that they have been given an unlucky fate or destiny.
From there, you should be able to see why they are star cross'd. They are unlucky in who they fall in love with. They are unlucky in Romeo being unable to leave the city like he is supposed to. They are unlucky in having him get to the tomb before Juliet wakes up.
So Romeo and Juliet are unlucky all the way around. That's why Shakespeare uses this figure of speech to describe them.