What does the line Let me not to the marriage of true minds admit impediments mean?

The phraseology evokes - almost certainly deliberately - Words from the Christian church marriage service. However, the relationship here is one between two persons who, when in their right ("true") minds have been closely in tune with each other. It is a friendship - albeit one which has been sufficiently strong to be compared to the affinity of a marriage.

The "impediments" imply that the friendship is threatened by the misbehavior of either the poet or his friend. Interestingly, the immediately following sonnets depict a poet trying to excuse his misdemeanors. With this flavor, the line in question becomes the introduction to a plea. "Don't", the poet appears to be saying, "let my behavior bring impediments to the continuation of our great friendship". He goes on to declare in beautiful, flowery language that true love (implicitly that which the friend should have for him) should overcome and outlast any obstacle.