The two most striking aspects of the punctuation are Dickinson's use of exclamation points (there are six) and dashes (eight).
The exclamation points emphasize the dramatic situation of the poem, in which the poet addresses the reader directly. The opening exclamation—"I'm Nobody!"—is a kind of confession, but the use of the exclamation point raises the tone, making this beginning seem a bit like a joke or prank, or perhaps just a knowing confidence between friends. This sense of sharing a secret is reinforced in the third line, in which the poet realizes that perhaps she and her reader are a "pair" of nobodies—here, the exclamation point registers excitement at the prospect. Then, in the next line, the exclamation points again emphasize the secrecy of this bond ("Don't tell!") and the informal way in which the reader is being addressed ("you know!")
Dashes indicate hesitation, but also a kind of unexplained logical connection. In the second line, the dashes that set off "Nobody" suggest that this is a word the poet perhaps is shy about using. In the fourth line, using the dash to set off "you know!" indicates a certain off-hand observation, the "you know" an excuse for not explaining what she means by "they'd advertise."
On the other hand, the dashes setting off "to be" in line five are difficult to parse. Leaving them out ("How dreary to be somebody!") definitely changes the tone of the line, which becomes an open assertion. The dashes give the line a tentative character. It is also possible to think of the dashes here as calling attention to the verb "to be," bringing the reader's attention to the central problem of the poem, which is existence.
It's also possible to think of the dashes as Dickinson's way of controlling rhythm. The "hitch" in meter the dash creates makes the sound of the poem more complex, and, especially in lines six and seven, suggests that the simile ("like a frog"), which on the surface seems meant to be a joke, might deserve more careful consideration.