A linear operator $A$ is said to be normal if $AA^\dagger = A^\dagger A$. The significance of the class of normal operators is that these are precisely the operators that have an orthonormal basis of eigenvectors. Consequently, if $A$ is normal then
A = \sum_k \lambda_k |\psi_k\rangle\langle\psi_k|\tag1
for some complex numbers $\lambda_k$ and orthonormal basis $|\psi_k\rangle$. It is easy to see that $\lambda_k$ are $A$'s eigenvalues and $|\psi_k\rangle$ are $A$'s eigenvectors associated with the respective eigenvalues $\lambda_k$. Nielsen & Chuang introduces normal operators on page 70 and proves the above theorem in box 2.2 on page 72.
All Hermitian and unitary operators are normal. Moreover, it is easy to check that the eigenvalues of a Hermitian operator are real. If in addition the operator is positive (i.e. positive semi-definite) then its eigenvalues are non-negative real numbers.
Now, the operator $A$ in equation $(2.172)$ is easily seen to be positive (and therefore Hermitian by definition). Thus, $A$ is normal and can be written as in $(1)$. Moreover, since $A$ is positive then $\lambda_k$ are non-negative real numbers.