In this scene of Romeo and Juliet, by William Shakespeare, Romeo and his new wife, Juliet, have just passed their first night together as husband and wife. Romeo, having killed Tybalt in response to Mercutio's murder at Tybalt's hand, has been banished from Verona. If he is caught within the city at the break of day, he will be killed.
Romeo has especially suffered from this punishment: separation from Juliet is worse, in his eyes, than death.
Ha! Banishment! Be merciful, say 'death.' / For exile hath more terror in his look, / Much more than death. Do not say 'banishment.' (III, iii, 13-15)
"More light and light—more dark and dark our woes" paraphrased means as the sun rises (more light), our sorrows are much greater (darkened). However, the meaning with regard to the young couple is that with every moment that draws them closer to dawn, Romeo's banishment must be fulfilled or he will die. In other words, as the day begins, their sorrows increase because he faces the threat of death and they will be separated.
Romeo says this in context while he and Juliet are trying to figure out if it is day or night after they have spent the evening together. When he says this particular statement, they have figured out that it is indeed light outside and that means it is indeed almost day and the watch will be set and Romeo is about to get busted if he stays with her. So, the "more light and light" portion refers to the time of day, and the "more dark and dark our woes" refers to the fact that their problems are going to get worse and worse the longer he stays. The connotation of dark in this sense is the idea of danger, trouble, or problem.
In this circumstance, the words are used both literally and metaphorically, with "dark" having a double metaphor: situational trouble/feelings of sorrow. Throughout the text of Romeo and Juliet, light and darkness are also regularly used to represent good and evil. Here, the growing light is tied to trouble for them. This is ironic.