Economic decline is probably the primary reason, but mixed with other factors like decolonisation and the new global balance of power.

It's easy to miss the economic impact of the two world wars on the UK because the long-term effect on living standards was not severe. Before WW1, the UK was the largest foreign investor in the world. The United States was the largest recipient. These investments were largely liquidated during the First World War. In the 20s and 30s although the country kept some of its strengths, eg. the largest merchant marine in the world, it was clearly losing ground to the US and to other countries. British shipyards and ship design were falling behind in terms of productivity and technology from at least the 1930s, and WW2 made things worse.

In the Second World War, the UK became a massive debtor to the United States (not to mention the "billion dollar gift" from Canada). Immediately following the war economic conditions were severe - rationing was worse post-war, and the UK had to go to the IMF for a loan. (Incidentally, the repayment of the main US loans was never a massive problem, but it came with a condition - convertibility for sterling by (iirc) 1948, which was.) Unlike the US, the UK was forced to spend a lot of money replacing capital and housing. The priority for government spending was social.

Economic conditions were a lot better by the 1960s, but by this time other things had changed, and the UK never regained its pre-war economic status.

The second factor is decolonisation. One of the reasons for the Royal Navy has been to seize, coerce and defend Imperial possessions. In WW2 it was unable to do this - the German surface fleet was defeated, but the Far East could not be held against the Japanese. After the war, the Empire was steadily dismantled. Britain did not have the military, economic, political or ideological resources to hold it (as shown by India, Suez, etc.). The USSR was actively in favour of decolonisation, while the US was not supportive (except where it perceived a serious socialist or communist threat). So the end of empire removed one of the core reasons for a powerful navy.

The third factor is: what wars would the navy be involved in? It was self-evident that the UK could not take on the USSR, so should a war break out, it would be supporting the United States. France and Germany developed into (fairly) firm allies rather than rivals. Third-world countries as yet had little naval power (and even up to the 1980s the Royal Navy greatly outclassed the Argentine Navy). Even the diminished fleet was likely to be superior to most plausible opponents.

On top of that, any new world conflict was likely to be a nuclear one, so the value of conventional forces was greatly reduced. The key strategic priority was air defence and nuclear bombers - and later nuclear missiles and submarines. In terms of pure destructive power, trident alone puts the British Navy near the top of the world's navies.

I would argue that postwar, the significance of the Navy to British identity (or identities, if you prefer) has diminished. The Air Force took much of its prestige. The myth most remembered from WW2 is the Battle of Britain, not the Battle of the Atlantic (though this was not the case in the 1950s), and aircraft design was one area where the UK had something of an edge. In the last couple of decades the Army has become more prominent, particularly following its deployment in Iraq and Afghanistan.

(Should be able to dig out sources for at least the economic stuff in here if anyone wants.)

They didn't give up on it, in fact postwar governments practically bankrupted the county trying to maintain the British military together with a policy knows as 'East of Suez' where Britain attempted to maintain it's influence through the world.

When the rest of Europe was building road and rail networks that were the envy of the world, the UK was sinking money into money pit projects such as the V bombers.

Correlli Barnett wrote extensively about this in a series of books, they get a bit repetitive but the final book in his series 'The Verdict of Peace' is worth reading by itself, and details exactly how the UK committed economic suicide after the war.