I think it is somewhat understandable for small territories that are either still dependent on the United Kingdom or are independent but relying economically on the UK, but what about Canada, Belize, Australia, New Zealand and Papua New Guinea? Why do they keep Queen Elizabeth II as their head of state?
In part this is tied to the notion of the Commonwealth of Nations. Unlike the US, nations of the Commonwealth gained independence through civil and legal procedures as the British colonial system dissolved. Those nations with long, favorable histories as part of the Empire — like Canada and Australia — largely opted to retain the Queen and the constitutional monarchy system. Other nations — mainly those which were primarily economic colonies, not settlement areas — took on a monarchy-independent republican form of government.
Not all countries are the same, and may have different reasons. The reasons given for Australia in this answer are likely to be common across other countries. To summarize:
For a lot of people having a Royal family is fun, people like reading about them in magazines, and in that way they're like celebrities.
Some people see the Royal family and the links to Britain, as an important part of the country's heritage. "if it ain't broke don't fix it."
Countries other than the UK don't pay anything for the Royal family.
The monarchy has the ability to remove a government if it's abusing its power and that way they're like an extra safety net for our democracy.
The Westminster system has been exported to many countries with success and is no less 'democratic' than Republican models. In the last four years this argument has been even more effective.
Republicanism would be more expensive than a monarchy.
There are benefits to having a head of state who is not a politician.
You don't fix something unless it is broken.
There are enough broken things to be fixed.
Disclosure: I am a conservative, with conservative friends in AU, NZ, and CA. In some years we are the majority of those countries, and we take extra caution before changing something, unless it is proven to be broken.
The resource is limited and we need to focus more on other things that are clearly broken, for example the inflation, job-loss, etc
As a liberal, Australian PM holds similar beliefs.