Why measure pitch at all?
Because ears are unreliable narrators. A voice that feels higher today may not be, and a change too gradual to hear is still a change. Pitch measured in hertz gives a therapist and a patient the same reference to talk about — and a record that survives between appointments.
The useful figures are rarely a single reading. Minimum, maximum, range and average across a recording say far more than one number, because the interesting thing is usually how much the voice moves rather than where it sits.
What is a normal pitch?
There is no single answer, and treating one as normal is unhelpful. Adult speaking pitch varies widely with anatomy, age, language and context, and therapy goals are usually set relative to a person's own baseline rather than to a population figure. What matters is the direction of travel and whether the voice is comfortable and sustainable.
Can this be done at home?
Yes, that is exactly the point. A recording made at home between sessions captures the voice as it is actually used, rather than as it performs in a clinic room. Recordings and their statistics can be sent to a therapist, which integrates speech therapy into everyday life situations. You can also do this independently, without support from a speech and language therapist.