Finding Kp using log Kp
The $K_p$'s here are those for the standard formation reactions.
If you go to the following link, you will see $\Delta G^o_f$ values for $\ce{CO2(g)}$  and $\ce{CO(g)}$, in $100 \,\mathrm{K}$ increments, up to $1500 \,\mathrm{K}$:
https://www.google.com/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=&amp;ved=2ahUKEwjbt56Z9IbvAhXVjp4KHbD-BS0QFjAJegQIAhAD&amp;url=https%3A%2F%2Flink.springer.com%2Fcontent%2Fpdf%2Fbbm%253A978-3-319-41616-8%252F1.pdf&amp;usg=AOvVaw3rJgF0nzpqXtq9ML9qvGaV
Thus, doing a simple linear extrapolation to $1600 \,\mathrm{K}$ gives us the following:
$$\Delta G^o_f,\ce{CO_2(g)} = -396.309 \mathrm{\frac{kJ}{mol}}$$
$$\Delta G^o_f, \ce{CO(g)} =  -252.3298 \mathrm{\frac{kJ}{mol}}$$
Solving for $\ln K_p$ (using $\Delta G^o = -RT \ln K_p$) gives:
$$\ln K_p,\ce{CO_2(g)} = 29.7906$$
$$\ln K_p, \ce{CO(g)} = 18.9677$$
Dividing by $\ln(10)$ to convert from the natural logarithm to the base-10 logarithm gives:
$$\log K_p,\ce{CO_2(g)} = 12.940$$
$$\log K_p, \ce{CO(g)} = 8.238$$
[$\Delta G^o_f$ for $\ce{CO(g)}$ changes more rapidly with temperature than that for $\ce{CO2(g)}$, so my extrapolation for the former may have a larger error, which could explain the small discrepancy vs. the value you were given.]
In addition, if you appropriately combine the formation reactions to give the net reaction (as you would for a Hess's Law problem), you will see that you get the supplied equation for $\log K_p$ for the net reaction (I'll leave this as an exercise for you).