“make an appointment” / “come in”
An appointment is an arranged time to meet a person, such as a doctor, hairdresser, or interviewer. A booking for a restaurant table or hotel room is a reservation, as in A1-50. Make an appointment means to arrange the meeting. Here come in means to visit the clinic, so When would you like to come in? asks for the desired appointment time.
“And what seems to be the problem?”
What seems to be the problem? is a standard way for medical staff or service staff to ask why someone needs help. Seems to be makes it gentler and more polite than the direct What's the problem?
“stitches” / “new thread” / “pink thread”
Stitches are the threads or seams used to sew up a wound, and came loose means they have come undone. Thread is the material used for sewing. Because Frank is an artificial human, he calmly explains that a loose stitch on his arm just needs new thread. The punchline is that Tina cares more about the thread's color than the treatment and asks for her favorite, pink.