It's only part of the story but it's easy to find a justification for a Common Fisheries Policy: Fish moves around and depletion of the resource impacts everybody. So much so that there are in fact other international organisations dealing with this issue to which the EU defers in part when setting its “total allowable catch”. Beyond that, the story is somewhat similar to the Common Agricultural Policy: It was there from the start (and as such a “take it or leave it” proposition for subsequent would-be members) and is sustained by powerful constituencies.
Besides, it's not just the “fish of the UK EEZ” that is being shared and the UK hasn't “relinquished” anything. The regulation and resources are pooled, which is not quite the same. The sector-by-sector zero-sum view of EU policies that is so common in the UK is short-sighted but in this case the UK might be a net beneficiary: UK quotas are the second largest in the EU and their overall share of the total allowable catch is larger than what is being caught in the UK EEZ. As always with trade, what's really at stake are competing local interests (big boats vs. large boats, specific fisheries, supermarkets, consumers…) and long-term sustainability rather than the interests of the country as a whole.