This phrase is useful for scheduling meetings and appointments. When you're talking to someone with a busy schedule, you may have to ask them to "fit you in". "Fitting you in" means fitting you into their schedule. You can imagine a schedule book with each meeting or appointment taking up a block of time. If the person has enough room to add your appointment between two other appointments, then you say that they could "fit you in".
In the example above, the speaker asks the receptionist if they can fit her in, because usually getting a haircut requires you to make an appointment a week or a few days earlier.
The receptionist might answer by saying:
Let me see... I can fit you in from 4:30 to 5:00. Would that work for you?