Why does a Turing Machine need at least two states?
TL;DR: technically, there need to be an accepting and rejecting state and they need to be different; but it's not a big deal and, at an intuitive level, one-state machines work just gine.



A Turing machine is typically defined to be a tuple $M=\langle Q, \Sigma, \Gamma, \delta, q_0, q_\mathrm{acc}, q_\mathrm{rej}\rangle$ where the items are the state set, input and tape alphabets, the transition function (or relation for a nondeterministic machine), and the initial, accepting and rejecting states, respectively.

The accepting and rejecting states must be distinct because the machine cannot simultaneously accept and reject. Therefore, there must be at least two states.

In principle, you might want to define a Turing machine that has only one state: the initial state would be the accepting state and there'd be no rejecting state, or vice-versa. There's nothing morally wrong with that but it would make the definition more complicated.  Instead of saying "A Turing machine has distinct accepting and rejecting states", you'd have to say "A Turing machine either has distinct accepting and rejecting states, or it has just one state which is either the accepting state or the rejecting state." That's more complex, and it doesn't buy you anything, so people don't do it.

But note that this isn't really an important point. You can perfectly well think of Turing machines with just one state and it won't hurt you in any way, unless you try to write down a formal definition. Any machine that you want to think of as having one state, you can easily rewrite as having two states: just add the second state, which will be inaccessible.