I read that the market will find the equilibrium how much these transaction fees will be.
It will not. This is perhaps the biggest flaw in Bitcoin at the moment: once mining rewards end there is no direct linkage between the amount of hashpower needed to secure the network and the incentive to mine.
True, there is a limit on the blocksize, so if the transaction volume in a block window (approximately 10 minutes) exceeds the block size you can expect a miniature "auction" where transactions fight for space in the block by bidding up the minimum transaction fee needed to get in. However this isn't really a closed-loop adjustment: the maximum blocksize is an arbitrarily chosen number, and there's no reason to believe the maximum blocksize is small enough to ensure that transaction fees are high enough to incent enough miners to mine to keep the system secure. Unlike the difficulty and the USD/BTC exchange rate it does not respond to market activity. It also has the negative side effect of capping the worldwide Bitcoin transaction throughput since other parts of the protocol rely on the assumption that blocks are created -- in the long run -- no more than once every ten minutes.
Compare this to the current situation with mining rewards: the more valuable a bitcoin is the more incentive there is for somebody to try to overwhelm the "good guys" by gaining 50%+1 hashpower. However, the more valuable a bitcoin is the more miners will mine! It isn't perfect, but the important point is that the demand for security increases the incentive to mine. Note that although the difficulty will go up, that simply ensures that the reward granted every ten minutes is an approximately constant number of BTC -- the number of terahashes/sec fighting over that amount of BTC is free to respond to changes in their changing value (as measured in terms of all other goods in the world, including other currencies).
As the mining reward is reduced this "direct coupling" between the network's need for security and the incentive to mine becomes progressively more diluted.
I worry a lot about what will happen to Bitcoin once we decouple those two forces. I think the developers ought to at least come up with a story on how this will be solved so people can start testing it.