In the play "Romeo and Juliet" by William Shakespeare, the author tells us in advance that Romeo and Juliet's love (or material welfare together in the future) is doomed from the start (star-cross'd.) For example, no matter how much he might love his sweetheart - his family never will because they have an irrational hatred of her family. Therefore, they cannot swear the usual vows - there will be no dowry/inheritance for either of them, therefore no likely property or children to found a new family upon. The vows of love/faithfulness might be there and sworn to - but there is nothing concrete to back them up without the support of their parents.
Each word of this statement carries such weight. The word breathe refers to the abilities to utter, speak, or discuss. Vows refers to promises or commitments. It is also often and only used in engagements or weddings. This foreshadows what is about to come in the rest of the act as access will be difficult. However, the full quote uses the words "may not" which leaves the chance open that they might get to speak.
This phrase refers to the fact that Romeo can't articulate or verbally express his love, almost insinuating that he will do the opposite, act on it. Watch for it.
The reason that Romeo cannot have access to do this is because of the feud that exists between his family and Juliet's.
The meaning of the lines you cite is that Romeo cannot go to Juliet openly. He cannot go to her and talk to her of love. He cannot woo her and marry her the way that lovers might normally do.
The Prologue here is reminding us that the two lovers cannot be together in the normal way. But it is also telling us that in the next act, they will find ways to be together.
When they do, the stage is set for the tragedy.