The miners then choose the transactions they want to include in a way, as to maximize their gains. As a block can only hold a certain amount of transactions and the limit being set by the weight-units [1], miners will want to prioritize the transactions that offer them bigger fees per weight unit, or usually expressed as sats/vbyte (satoshis per virtual bytes).
So if you don't need a transaction to arrive in the next 1-2 hours, you can lower your fees, and the transaction might pass in the night (although BTC is international, congestion is less on US night time and especially week-ends) or if not, probably in the week-end. If it doesn't clear fast enough, or as fast as hoped, then RBF (replace-by-fee) or CPFP (child-pays-for-parent) might help (manually or an increase of the fee could be automated by the wallet every set amount of time).
TL;DR: there is a free market for transactions, but the price is not dictated by the market (and there is no real bidding system), instead you suggest a fee, and see if any miner around the world wants to include it in a block (and you can adjust if the outcome isn't achieved, which has similar results to a bidding system... but is more like natural supply/demand mechanism, as miner do not guarantee to include a transaction).
Helpful to see current unconfirmed transactions (and sats/vbyte): https://jochen-hoenicke.de/queue/#0,30d [info: the legend is clickable, and makes it easier to view how many higher paying transactions there are]
It turns out that people were right when they said that it would take a long while for it to be adopted.