In the complex $[FeF_6]^{3-}$, the central metal ion is Iron (Fe) in the +3 oxidation state. The electronic configuration of $Fe^{3+}$ is $3d^5$.

1. Since the fluoride ion ($F^-$) is a weak field ligand, the crystal field splitting energy ($\Delta_o$) is less than the pairing energy ($P$). Therefore, the pairing of electrons does not occur, and it forms a high-spin complex.
2. The distribution of 5 electrons in the $d$-orbitals will be $t_{2g}^3 e_g^2$.
3. Calculation of CFSE: $CFSE = (-0.4 \times n_{t_{2g}} + 0.6 \times n_{e_g}) \Delta_o$ $CFSE = (-0.4 \times 3 + 0.6 \times 2) \Delta_o = (-1.2 + 1.2) \Delta_o = 0 \Delta_o$.
4. Calculation of Magnetic Moment: The number of unpaired electrons ($n$) in $t_{2g}^3 e_g^2$ is 5. The spin-only magnetic moment ($\mu$) is given by $\mu = \sqrt{n(n+2)}$ BM. $\mu = \sqrt{5(5+2)} = \sqrt{35} \approx 5.92$ BM.

Therefore, the CFSE is $0 \Delta_o$ and the spin-only magnetic moment is 5.9 BM.