how can it be a separate thing if it has been a part of the state since its inception?
The Above State with its Presidents and Secretaries and Legislators are all replacable and expendable, they're just an abstraction and obfuscation layer for the Under State to put their desires into action.
Occasionally you might catch glimpses of them when they play their hand too strongly, like the Iraqi Weapons of Mass Destruction(tm) narrative pushed by the Intelligence Community (the Under State) with State Secretary Colin Powell (the Above State) acting as their mouthpiece. On rare occasions a Deep State agent might turn coat and blow the secrets as was the case with Snowden.
i've been in a cult my entire life (to give you an idea i live in a house full of fanatics rn) so the thought of voting or having a say seem very alien to me
why do people care so much about that?
Because governments are usually accountable to someone or something.
For example, in most monarchies it's royal blood or even a god. In democracies like the US it's the voters.
Deep States are more powerful than governments (they are the power) and not accountable to anyone and arguably not even desired by the people governed, that is why people care.
Let's rename Deep State as "Under State" for the purposes of this conversation to make it easier to understand.
Whatever is the government (the "State", like in "Head of State") you can see is the "Above State". Presidents, Prime Ministers, Kings, Governors, Legislatures, Courts, the government that supposedly runs the country, this is the Above State.
The Deep State, the Under State, operates under the Above State for their own goals and benefits accountable to noone, the only time they come up to the sunlight is to correct an Above State that dares to harm or defy the Under State. You can almost never see the Under State as a common citizen, but you can probably feel their influence if you pay attention.
"Deep" in Deep State hence refers to where they are figuratively located in a given country's political mechanism.