1) They sit at the apex of a pyramid of ranks, precedences and privileges.
https://en.wikipedia.org/wiki/Order_of_precedence_in_England_and_Wales
"Your Majesty" outranks "Your Royal Highness" which outranks "Your Highness" which outranks "Your Grace" which outranks "My Lord", and so on and so forth, descending through all the various tiers of Knighthoods, (diverting on the way for people like the Lord Lieutenants) until you get to the bottom layer of ordinary citizens.
Depending on which layer you're born into (or appointed to by greasing palms or helping politicians in various ways) you end up having disproportionate influence, or actual unelected legislative power.
Traditionally, the British nobility rank directly below the British royal family. In the modern era, this ranking is more of a formally recognised social dignity, rather than something conveying practical authority; however, through bodies such as the House of Lords, the nature of some offices in the Royal Household, and British property law, the British nobility retain some aspects of political and legal power.
Did you know that there are only two countries in the world which reserve seats in their legislatures for unelected representatives of the state religion?
Iran and the UK.
2) They secretly and privately influence our "democratically elected" governments to act in their own personal interests.
New documents expose a chink in the queen’s armour of secrecy.
theconversation.com
Government documents that “relate to” communications with the sovereign or the next two persons in line to the throne, as well as palace officials acting on their behalf, are subject to an
absolute exemption from release under freedom of information or by government archives.
3) Governments seeking to impose their will on our democratically elected representatives in Parliament will use the same out-of-band channels to get the monarch to act in
their own partisan interests.
And I'm sure we all remember this:
en.wikipedia.org