How do television networks determine how many viewers they have for each show?

Cable TV knows which show you are watching, and they report it.

Neilson ratings mostly.   They take reports from a large sample pool and pair those  figures with algorithms based on estimates of what demographics and population percentages are typically watching TV at those dates and times.

Neilson rating. 

If you participate they give you a form to mark what you watch and then submit by phone, mail, and now probably online. 

They invited me once but I don't watch TV.