How do we use oxygen isotopes as temperature proxy?
This is somewhat a general answer.  According to the Royal Chemical Society webpage Chemical climate proxies (Evans, 2013), the rarer and heavier $\ce{^18O}$ isotope in water generally evaporates after and condenses (and precipitates) out earlier than the lighter and more common $\ce{^16O}$ isotope, shown in the diagram below:



Source: University of Michigan's page Past Climates on Earth

The colder the environment, the quicker the atmospheric water vapour condenses, hence the quicker the $\ce{^18O}$ isotope is depleted, giving a lower ratio of $\ce{^18O}$ to $\ce{^16O}$ (referred to by the RSC page as $\delta{\ce{^18O}}$) by the time it reaches the poles.  In contrast, a warmer environment would mean that the water vapour condenses later and slower, meaning that less $\ce{^18O}$ laden water vapour condenses, resulting in a higher $\delta{\ce{^18O}}$ value.

In short, it is not so much of the amount of $\ce{^16O}$, but rather the ratio of $\ce{^18O}$ to $\ce{^16O}$ isotopes present, otherwise known as $\delta{\ce{^18O}}$.

The following Earth Observatory diagram demonstrates the present-time process, showing the differences between warm and cold environments:



Source: Earth Observatory's page: Paleoclimatology: The Oxygen Balance