Geminis are all about talking and communcation and they're an extroverted air sign.
Thinking that your sun sign is the most important thing in your astrological chart is a common misconception, though.
Even I, a Sagittarius with a Sagittarius Ascendent and a Leo Moon, all extroverted fire signs, am still an introverted person!
When I first got into astrology, I didn't know why that was, either.
To explain to you what determines your introversion/extroversion, here's an example of two Geminis:
Heidi Klum and Johnny Depp.
She is so bubbly, always talking really quickly and really high-pitched.
Johnny Depp is a Gemini as well but he doesnt talk that much at all!
And if he isn't acting, he's usually a bit of a reluctant talker and doesnt seem to enjoy himself that much at public events.
The thing you have to look at when determining whether or why someone is an introvert or an extrovert is the dominant polarity of the signs your planets are in.
Might sound a bit complicated, but it really isnt.
Just take your complete chart and look at the signs of your ascendent and the ten planets and look at what polarity outweighs the other one.
Extroverted Signs = Positive Polarity
All signs of the fire element and the air element.
Aries, Leo, Sagittarius, Gemini, Libra, Aquarius
Introverted Signs = Negative Polarity
All signs of the earth element and the water element.
Taurus, Virgo, Capricorn, Cancer, Scorpio, Pisces
Now just count them together!
As for the two Gemini examples:
Heidis dominant signs are Gemini and Leo, both of positive polarity.
Johnnys dominant signs, on the other hand, are Taurus and Virgo, both of negative polarity.
I hope this helped!