When you are the owner/user and not the landlord you should be paying the taxes because most of the taxes are for use in the local community and most of that is schooling. The landlord gets no benefit from the majority of that stuff. The tenant does.
How does he benefit from the rental property? He doesn't live there an use the services. If he lives somewhere else in the town he pays taxes based on that. He charges rent for the other location to cover the taxes because he doesn't benefit from those taxes. The renter does.
But he passes the cost of those taxes on to the renter because the renter is the one benefiting from the schools/roads/police/fire/etc that the taxes fund.
That is often true. A vacant rental still owes the taxes though. I can only say my time as a landlord I have not always covered those costs with rental income. Sometimes the property is empty or doesn't command enough income to cover all your costs.