The context is as follows.
The buffer is now unpinned and is a candidate for immediate aging out, if the current contents (data block) are not referenced again.
This seems more like a programming/computer related question to me, due to the feel of the website you got the quote from.
However, it seems that the question is mostly answered by another quote from the same site:
The Oracle server uses the least recently used algorithm to age out buffers that have not been accessed recently to make room for new blocks in the database buffer cache.
This means that it retires those buffers from use; though, since I'm not a programmer, I don't know what happens to them after that.
As reported in the Wikipedia, aging out is "American popular culture vernacular used to describe anytime a youth leaves a formal system of care designed to provide services below a certain age level."
In the sentence you report, the phrasal verb is used figuratively. It means the buffer is released, and the memory it used returns in the free memory pool.
To "age out" in general is to become ineligible or irrelevant by virtue of physical age.
It's used in many "junior" sports and competitions that are tied more to physical age than academic grade level. An example is DCI marching competitions; most drum corps are open to individuals from high school through college-age, but on a person's 25th birthday they are no longer eligible to compete; they have "aged out".
It's also used in certain other situations, such as government programs (you can "age out" of eligibility for health insurance coverage under your parents in the U.S.), non-profits, etc etc.