Back in the Medieval England, cities had royal charters giving them rights. Someone who had the Freedom of the City could exercise the rights set out in the city charter, which generally meant they could hold property, they could engage in trade, they could elect city officials, they were protected from arbitrary arrest, and so on. Eventually in English history, of course, these rights would be extended to everybody, so now being named to the freedom of a city is pretty much just honorary, like being given the key to the city.
As a supplement to Captain Amazing’s answer, it’s worth pointing out that Eastbourne was not incorporated as a borough until 1883 and therefore would never have had any non-honorary freemen.
The borough council’s own website has details of the legal definition of an honorary freeman, as well as a complete list of recipients.