The "plank of Reason" that is breaking is the speaker's grasp on sanity. The metaphor, then, is of the tenuous hold she has had on its retention. A plank is a narrow board, often suspended over water. When the plank breaks (reason, sanity) the speaker feels that she has plunged into the abyss.
The plank too, is a metaphor for crossing from one place (or state) to another. There are two "shores," if you will, in this poem. Sanity and insanity, life and death,and perhaps even another, order and disorder. It is the metaphor of the plank that has carried the speaker thus far. When it breaks, she is delivered from sanity, life, and order to insanity, death, and disorder.
By the way, anthologies often leave out this final stanza. Here are those last lines:
And then a Plank in Reason, broke,
And I dropped down and down
And hit the world at every plunge,
And finished, knowing --then --
I agree with the previous answer; it had to do with the loss of sanity associated with death, but Dickenson's religous roots also play a part. She came from a strong Calvinistic background and spent much of her adult life avoiding her religous upbringing. She was a great admirer of Emerson, a very popular rationalistic theologian. When she says "the plank of reason broke" I believe she is refering to her need to look further than Calvinism for her spiritual answers, especially at the time of death. It is one thing to declare an idea dead or of no meaning, but it is quite another to step out on a plank and trust her "reason" for her eternal salvation.