The speaker says that “They”—meaning the people around her and/or society as a whole—have attempted to “shut [her] up,” or imprison her, “in Prose.” Prose is any writing that is not poetry. Here, the speaker uses “Prose” as a metaphor for all of the boring, confining, and “prosaic” conventions of society. The speaker means that she has been “shut up” or confined by these conventions.
To understand what this means for the poem, it can be helpful to remember that the poet, Emily Dickinson, was a woman living in 19th-century New England. Women’s roles were limited and tightly circumscribed in this world: they would be expected to marry and have children, and to be quiet and obedient.
The “Prose” of social conventions, then, likely stands for all of these codes of behavior. These conventions would attempt to “shut” the speaker “up” literally, in limiting what she could do with her life, and figuratively, in the sense of silencing her. (As it happens, Dickinson did not obey these conventions: she never married, and in her poetry, she subverted a range of cultural norms throughout her life!)
The speaker, then, makes it clear from the outset that she has experienced confinement and constraint—or at least, attempts to confine her. She associates this confinement with societal conventions, and by extension with societal codes of behavior for women at the time Dickinson was writing.
Interestingly, though, the speaker also associates this sense of restriction and limitation with prose writing itself, with all of the conventions of syntax and punctuation that go along with it. The speaker implies that she could never escape the restrictive conventions of daily life by writing in prose; she can only do so through poetry.