One usually starts from the CCR for the creation/annihilation operators and derives from there the commutation rules for the fields.
However, one can start from either (see for example here about this).
Suppose we want then to start from the equal-time anticommutation rules for a Dirac field $\psi_\alpha(x)$:
$$ \tag{1} \{ \psi_\alpha(\textbf{x}), \psi_\beta^\dagger(\textbf{y}) \} = \delta_{\alpha \beta} \delta^3(\textbf{x}-\textbf{y}),$$
where $\psi_\alpha(x)$ has an expansion of the form
$$ \tag{2} \psi_\alpha(x) = \int \frac{d^3 p} {(2\pi)^3 2E_\textbf{p}} \sum_s\left\{ c_s(p) [u_s(p)]_\alpha e^{-ipx} + d_s^\dagger(p) [v_s(p)]_\alpha e^{ipx} \right\}$$
or more concisely
$$ \psi(x) = \int d\tilde{p} \left( c_p u_p e^{-ipx} + d_p^\dagger v_p e^{ipx} \right), $$
and we want to derive the CCR for the creation/annihilation operators:
$$ \tag{3} \{ a_s(p), a_{s'}^\dagger(q) \} = (2\pi)^3 (2 E_p) \delta_{s s'}\delta^3(\textbf{p}-\textbf{q}).$$
To do this, we want to express $a_s(p)$ in terms of $\psi(x)$. We have:
$$ \tag{4} a_s(\textbf{k}) = i \bar{u}_s(\textbf{k}) \int d^3 x \left[ e^{ikx} \partial_0 \psi(x) - \psi(x) \partial_0 e^{ikx} \right]\\
= i \bar{u}_s(\textbf{k}) \int d^3 x \,\, e^{ikx} \overset{\leftrightarrow}{\partial_0} \psi(x) $$
$$ \tag{5} a_s^\dagger (\textbf{k}) = -i \bar{u}_s(\textbf{k}) \int d^3 x \left[ e^{-ikx} \partial_0 \psi(x) - \psi(x) \partial_0 e^{-ikx} \right] \\
=-i \bar{u}_s(\textbf{k}) \int d^3 x \,\, e^{-ikx} \overset{\leftrightarrow}{\partial_0} \psi(x) $$
which you can verify by pulling the expansion (2) into (4) and (5).
Note that these hold for any $x_0$ on the RHS.
Now you just have to insert in the anticommutator on the LHS of (3) these expressions and use (1) (I can expand a little on this calculation if you need it).
most sources simply 'pull the $u, u^\dagger$ out of the commutators' to get (anti)commutators of only the creation/annihilation operators. How is this justified?
There is a big difference between a polarization spinor $u$ and a creation/destruction operator $c,c^\dagger$.
For fixed polarization $s$ and momentum $\textbf{p}$, $u_s(\textbf{p})$ is a four-component spinor, meaning that $u_s(\textbf{p})_\alpha \in \mathbb{C}$ for each $\alpha=1,2,3,4$.
Conversely, for fixed polarization $s$ and momentum $\textbf{p}$, $c_s(\textbf{p})$ is an operator in the Fock space. Not just a number, which makes meaningful wondering about (anti)commutators.
Best Answer
For starters, you can think of $\hat \psi$ as a set of operator fields instead of just one operator field (like $\hat \phi$). This is analogous to how a vector is different from a scalar. The set of $\hat\psi$ operators rotate into one another under transformation (like rotations), whereas the $\hat\phi$ just stays as it is.
In addition, the $\hat \phi$ and $\hat \psi$ usually describe particles of different statistics (bosons vs fermions, respectively). This is reflected in the commutation vs anti-commutation properties of the partial creation/annihilation operators.
In terms of path integrals, the "classical" $\psi$ fields are anti-commuting number fields, whereas the classical $\phi$ fields are regular numbers.