When does phenolphthalein containing solution change colour?
I am not absolutely sure I understood the question completely, but I will try to shed some light on the colour change of an indicator.

As a rule of thumb the human eye can make out a change in colour when there is a 10:1 ratio of the components. Since phenolphtalein, $\ce{HIn}$, is without colour, and its deprotonated form $\ce{In-}$ is pink, it means that a solution with


$\displaystyle \frac{n(\ce{HIn})}{n(\ce{In-})} = \frac{10}{1}$ will appear clear, while
$\displaystyle \frac{n(\ce{HIn})}{n(\ce{In-})} = \frac{1}{10}$ will appear pink.


Since phenolphtalein is a weak acid, you will see, that the colour change will theoretically happen around equilibrium constant, i.e. $\mathrm{pH}=\mathrm{p}K_\mathrm{a}$. (Which is also a simplification, because phenolphtalein has multiple acidic protons.) At that point you will probably not notice significant changes. Though phenolphtalein is a bit of an exception here, because it has only one colour. The human eye can probably see a smaller surplus of the colourful component. (It is much more difficult for an indicator like methyl red.)

If you substitute the above fractions into the Henderson-Hasselbalch equation, you can approximate the range in which the colour change will happen:
\begin{align}
\mathrm{pH} 
  &amp;= \mathrm{p}K_\mathrm{a} - \lg\left(\frac{n(\ce{HIn})}{n(\ce{In-})}\right)\\
  &amp;=%
  \begin{cases}
     \mathrm{p}K_\mathrm{a} - \lg\left(\frac{10}{1}\right); &amp; \text{clear}\\
     \mathrm{p}K_\mathrm{a} - \lg\left(\frac{1}{10}\right); &amp; \text{pink}  \end{cases}\\
\mathrm{pH} 
  &amp;= \mathrm{p}K_\mathrm{a} \pm 1
\end{align}



When you are estimating the amount of carbon dioxide dissolved in water, then the following equilibrium will be present:
$$\ce{CO2 + H2O &lt;=&gt; HCO3- + H+ &lt;=&gt; H2CO3}$$

Adding hydroxide ions you are shifting this equilibrium to the right, essentially deprotonating the acid. Once you have fully (in first approximation) converted it to carbonate, you start deprotonating the indicator until you notice the colour change.