Either people will pay more, in which case landlords will raise their rents until people stop being willing to pay more, or people won't pay more, no matter what the cost to the landlord.
Nowhere in this does the landlord show up with a tax document and say "see folks, now you gotta pay MOAR TAX".
Similarly, lowering the tax rate would provide greater profit to landlords, which would cause new entrants to the rental market, leading to greater supply and a lower average cost.
That's just the basic theory of competitive markets, at least as I remember it. It starts with the 'X' shaped chart of supply vs demand which you draw over and over in Microeconomics 101. Macro takes the concepts from Micro and connects the dots to get from increasing taxes -> higher rental prices.
The reasoning is multi-part:
1. It aligns tenant and landords by encouraging renters to vote against property tax increases. Renters can now think whether the services increase is worth the tax increase instead of only worrying about the benefit side as the cost side is 100% someone else problem absent an escalator.
2. It allows landlords to offer their property for rent without including a "holdback" for their guess as to their property tax increase.
Take an extreme example, if taxes were $5000/mo for a 800 sq ft 1BR then rent for a 1BR would be greater than $5000/mo. There might be very few units renting at that price and home prices would approach zero or could even become significantly negative if owners had no other way to escape the tax bill.
You see this for some properties in Detroit where the tax bill is more of a liability than the house and land are worth so the plot can be purchased for $1 for any sucker who will take it (or is willing to carry to tax liability in the hopes the asset will eventually appreciate beyond the accumulated taxes)
Speaking of Econ 101, when taxes shift the supply curve, what happens next? Some renters are priced out, others get less house for more money. Under no circumstance does some fat-cat middle man reach into their own pocket to help a brother out.