Trying to understand the proof of the halting problem presented in Sipser textbook
The first misconception is that $A_{TM}$ is a Turing Macahine. $A_{TM}$ isn't a machine, it's a language, $H$ is the machine that decides the language $A_{TM}$. So you give $H$ a string that consists of two parts, a string $M$ that describes a Turing machine, and another string $w$, which can be anything.

This leads us to the second part, $w$ can be any string - it's the theoretical input to $M$ that you want to know whether $M$ would accept or not. So $H$ should accept if $M$ would accept $w$, and reject otherwise (i.e. when $M$ doesn't accept, which could be by rejecting or never halting). $\langle M \rangle$ is just a string, it happens to coincidentally be a description of a Turing Machine, but it is still just a string. So there's no reason we can't ask "does $M$ accept the string which is its own description?"