When you invest in an S&P500 index fund that is priced in USD, the only major risk you bear is the risk associated with the equity that comprises the index, since both the equities and the index fund are priced in USD.
The fund in your question, however, is priced in EUR. For a fund like this to match the performance of the S&P500, which is priced in USD, as closely as possible, it needs to hedge against fluctuations in the EUR/USD exchange rate. If the fund simply converted EUR to USD then invested in an S&P500 index fund priced in USD, the EUR-priced fund may fail to match the USD-priced fund because of exchange rate fluctuations.
Why hedging is important
Here is a simple example demonstrating why hedging is necessary. I assumed the current value of the USD-priced S&P500 index fund is 1,600 USD/share. The exchange rate is 1.3 USD/EUR. If you purchase one share of this index using EUR, you would pay 1230.77 EUR/share:
If the S&P500 increases 10% to 1760 USD/share and the exchange rate remains unchanged, the value of the your investment in the EUR fund also increases by 10% (both sides of the equation are multiplied by 1.1):
However, the currency risk comes into play when the EUR/USD exchange rate changes. Take the 10% increase in the price of the USD index occurring in tandem with an appreciation of the EUR to 1.4 USD/EUR:
Although the USD-priced index gained 10%, the appreciation of the EUR means that the EUR value of your investment is almost unchanged from the first equation.
For investments priced in EUR that invest in securities priced in USD, the presence of this additional currency risk mandates the use of a hedge if the indexes are going to track. The fund you linked to uses swap contracts, which I discuss in detail below, to hedge against fluctuations in the EUR/USD exchange rate. Since these derivatives aren't free, the cost of the hedge is included in the expenses of the fund and may result in differences between the S&P500 Index and the S&P 500 Euro Hedged Index.
Also, it's important to realize that any time you invest in securities that are priced in a different currency than your own, you take on currency risk whether or not the investments aim to track indexes. This holds true even for securities that trade on an exchange in your local currency, like ADR's or GDR's. I wrote an answer that goes through a simple example in a similar fashion to the one above in that context, so you can read that for more information on currency risk in that context.
Process of hedging
There are several ways to investors, be they institutional or individual, can hedge against currency risk. iShares offers an ETF that tracks the S&P500 Euro Hedged Index and uses a over-the-counter currency swap contract called a month forward FX contract to hedge against the associated currency risk. In these contracts, two parties agree to swap some amount of one currency for another amount of another currency, at some time in the future. This allows both parties to effectively lock in an exchange rate for a given time period (a month in the case of the iShares ETF) and therefore protect themselves against exchange rate fluctuations in that period.
There are other forms of currency swaps, equity swaps, etc. that could be used to hedge against currency risk. In general, two parties agree to swap one quantity, like a EUR cash flow, payments of a fixed interest rate, etc. for another quantity, like a USD cash flow, payments based on a floating interest rate, etc. In many cases these are over-the-counter transactions, there isn't necessarily a standardized definition.
For example, if the European manager of a fund that tracks the S&P500 Euro Hedged Index is holding euros and wants to lock in an effective exchange rate of 1.4 USD/EUR (above the current exchange rate), he may find another party that is holding USD and wants to lock in the respective exchange rate of 0.71 EUR/USD. The other party could be an American fund manager that manages a USD-price fund that tracks the FTSE. By swapping USD and EUR, both parties can, at a price, lock in their desired exchange rates.
I want to clear up something else in your question too. It's not correct that the "S&P 500 is completely unrelated to the Euro." Far from it. There are many cases in which the EUR/USD exchange rate and the level of the S&P500 index could be related. For example:
Troublesome economic news in Europe could cause the euro to depreciate against the dollar as European investors flee to safety, e.g. invest in Treasury bills. However, this economic news could also cause US investors to feel that the global economy won't recover as soon as hoped, which could affect the S&P500.
If the euro appreciated against the dollar, for whatever reason, this could increase profits for US businesses that earn part of their profits in Europe. If a US company earns 1 million EUR and the exchange rate is 1.3 USD/EUR, the company earns 1.3 million USD. If the euro appreciates against the dollar to 1.4 USD/EUR in the next quarter and the company still earns 1 million EUR, they now earn 1.4 million USD. Even without additional sales, the US company earned a higher USD profit, which is reflected on their financial statements and could increase their share price (thus affecting the S&P500).
Combining examples 1 and 2, if a US company earns some of its profits in Europe and a recession hits in the EU, two things could happen simultaneously. A) The company's sales decline as European consumers scale back their spending, and B) the euro depreciates against the dollar as European investors sell euros and invest in safer securities denominated in other currencies (USD or not). The company suffers a loss in profits both from decreased sales and the depreciation of the EUR.
There are many more factors that could lead to correlation between the euro and the S&P500, or more generally, the European and American economies. The balance of trade, investor and consumer confidence, exposure of banks in one region to sovereign debt in another, the spread of asset/mortgage-backed securities from US financial firms to European banks, companies, municipalities, etc. all play a role.
One example of this last point comes from this article, which includes an interesting line:
Among the victims of America’s subprime crisis are eight municipalities in Norway, which lost a total of $125 million through subprime mortgage-related investments.
Long story short, these municipalities had mortgage-backed securities in their investment portfolios that were derived from, far down the line, subprime mortgages on US homes. I don't know the specific cities, but it really demonstrates how interconnected the world's economies are when an American family's payment on their subprime mortgage in, say, Chicago, can end up backing a derivative investment in the investment portfolio of, say, Hammerfest, Norway.