You are right that the bra and kets are conjugate transpose (or Hermitian conjugates) of each other. So, if |psi> is a column vector whose elements are w0 and w1, then <psi^dagger| is the row vector whose elements are w0* and w1* as shown on the bottom of page 187. That is, |psi>^dagger = <psi^dagger|.
The math notation can sometimes seem confusing. So, it’s best to understand the intent of what’s going on and then the notation will follow. Also, using actual numbers instead
of letters may help to clarify.