Why would people stare for a prolonged period of time and how do I stop feeling conscious?

Strange question I know. However, this has happened to me many times now with different people and its making me quite self conscious.
(I'm just a small petite girl)

One example is: I attended a meeting and everyone introduced themselves. I recognised one guy so when he introduced himself I said oh yeah I remember you from last week. He continued to stare for a while (he's quite a bit older than me) just wondering what the reason could be?
Why would people stare for a prolonged period of time and how do I stop feeling conscious?
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