In this poem, the speaker seems to be describing her feeling that she is losing her sense, her sanity, and that she is going crazy. She characterizes this loss of sense as a funeral in her brain with mourners who "Kept treading—treading," and a service that "Kept beating—beating"; the repetition of these actions seems to imply a kind of grating monotony, and this "numb[s]" her mind. The mourners "lift a Box" which is likely the coffin that bears her sanity, and then they "creak across [her] Soul / With those same Boots of Lead . . ." The sound of the heavy boots worn by the mourners tromping through her head echoes the repetitive treading of earlier in the poem, magnified now—for the reader—by the knowledge that the boots are leaden. They seem to symbolize the steady march of things that threaten to drive a person mad, or the things that a person thinks will ultimately make him or her go crazy. This could refer to the monotony of day-to-day life, the constant demands made on the individual by family, society, oneself. The repeated sounds in the poem make the speaker feel as if her "Sense were breaking through" and that her "mind was going numb"—both apparent descriptions of the loss of sanity and sense—and so the sound produced by the boots is certainly a partial cause of this.

In this poem, Emily Dickinson is describing a mental breakdown that is caused by the heavy despair and pain that the subject's  life has caused her to feel.  The funeral is for the sanity of the subject, as she sinks into depression.  

The lead boots are symbolic of the problems or disappointments of life that like, the heavy weight of lead trample on the subject to sink her into a a world of mental darkness. Her sorrows are so great that their heaviness weighs her down, she can bear no more.