How to prevent my introvert nature from being a barrier to success in grad school and future academic life?
If you experience severe anxiety communicating with other people, then you should consult a licensed mental health professional.  If anxiety is preventing you from succeeding in your PhD, then it is severe.
Practice can help you overcome your feelings when communicating.  Practice in an environment where you feel comfortable, and gradually your way to more challenging skills.  Good academics practice communicating all the time.
There are organizations which will facilitate practice, such as Toastmasters.  Your supervisor should also do that.