This means that every time you make a transaction you give 0.01 to the person who finds the next block.
Small correction. The fee is 0.01 BTC per 1KB of transaction size rounded up. A transaction with one input and two outputs (most typical) is 0.25 KB and the fee is 0.01. If your transaction has many inputs (like consolidating a wallet with many small change coins), the fee will be proportionally higher.
There is still no need to pay the fee unless you want to be (almost) sure to have the transaction included in the next block. The fee will probably become important when there are much more transactions going on and the competition for inclusion into the blocks is higher.
Kind of sounds stupid if Bitcoin gets really busy that you have to "pay fees" to make sure your transactions go by faster. This is a silly gimmick to the whole idea of Bitcoin doesn't make sense.