Why does the European Central Bank use the 5y5y rate to measure inflation I cannot understand why the 5y5y swap rate measures inflation. I can see how the floating rate somewhat measures the rate of interest rates which are related to inflation. However I cannot understand why this swap rate is the preferred way to measure (medium term) inflation expectations.

For more information see here.
In an interest rate swap, two parties swap with each other the obligation to pay a fixed rate for the obligation to pay a floating rate. Because it concerns a simple swap of obligations, the rates must have equal present value for both parties. The market quotes the fixed rate, so this is the variable of interest.

Draghi may have been referring to one of two types of swap contracts: the 5-year/5-year interest swap or the 5-year/5-year inflation rate swap. The case of the 5-year/5-year inflation rate swap is simplest. Because the contracts trade at equal present values, a lower fixed rate implies lower expected inflation.

The case for the interest swap requires an additional assumption. Consider that interest rates consist of two components: a compensation for inflation rate risk, and a compensation for default (and possibly other) risks. This latter part is called the risk premium, and it also affects the quoted fixed rates. So in order to draw any conclusions about market inflation expectations on the basis of interest rate swaps, it is simplest to assume that the risk premium is constant. Of course this is not necessarily the case, and this assumption could be investigated by looking at the difference between the two different swap contracts.

Finally, I cannot be certain about why the ECB prefers this measure to any other measure of inflation expectations. But besides the fact that it is a market measure, the timing adds up: the 5 year swap starts in 5 years, so it is an indicator of inflation 5-10 years in the future.