Whenever a Bitcoin transaction is made, it is broadcast to all nodes in the network.  All transactions since the previous block was solved are contained in the next block to be solved, and so all miners compete to be the first provide mathematical proof of work for the current block.

Let's say someone mines a block solo.  As we know, 50 BTC is generated with each solved block.  But let's say there was 1 transaction during that time containing a transaction fee of .005 BTC.  Then, that miner will receive 50.005 BTC.  Right now, the network is small and there are relatively few transactions, so the amount of transaction fees included in the block is small.

After all 21 million BTC have been generated, the 50 BTC will no longer be generated, but miners will still be rewarded with the transaction fees.  By this time, assuming that BTC is widely adopted, there will be many many times the number of transactions and thus the amount accumulated in transaction fees per block will be greatly increased.  It's possible that by this time, miners who solve a block solo will receive even more than 50 BTC, but they won't be new coins.  They will simply be existing coins that were included in the block in the form of transaction fees.