The poem uses caesura to create an atmosphere of misery, confusion, and oppression. Line 1 barely gets off the ground before it is interrupted by a dramatic exclamation mark:
My mother groan'd! || my father wept.
This divides the line in half (creating two parallel phrases), and means readers might linger on the word "groan'd" a beat longer, imagining a long moan of pain. By disturbing the poem's flow before it's even gotten started, this caesura also subtly reinforces the idea that the world will likewise restrict the flow of the speaker's energy.
Soon after, line 3 also uses a double whammy of comma caesura. Here, the speaker describes themselves as:
Helpless, || naked, || piping loud;
These commas compress the line, packing it with three distinct descriptions that capture the speaker's confused, powerless state. Even though caesurae often slow a line down, here they quicken the pace, putting the reader right there in the room with the screaming newborn baby.
It's worth noting, too, how these commas work hand in hand with asyndeton (the lack of a conjunction word like "and"). It's as though the speaker is gathering their first impressions of life, perceiving their own helplessness, nakedness, and noisiness in dizzying real time.