The woman is feeling incompetent. She knows the baby needs cleaning and changing (because he pooped), but she's saying she doesn't know the right way of doing it. When she says, "He's got extra bits," she means that the baby, being male, has a penis and testicles, and she's squeamish (uncomfortable) and unsure of how to go about it.
We know what needs cleaning is poop because the man subsequently says, "It's a little poop, nothing more."
The woman explains that there are no diapers in the house. The implication is that this is her fault. I infer that she has primary custody and has asked the man (who is the baby's father, and who doesn't live with her) to come over and help her figure out how to do basic care of a baby boy.
I think what happens at the end is that the man snorts and says, "I swear," under his breath.
(The baby's male anatomy is perfectly normal. The woman's squeamishness is probably a plot device.)