How to grow as a researcher outside of research? I am a Ph.D. student (after mid stage) working in theory. I do as much hard work as I can, but there is something I think I am missing. My experience so far is many people don't know what I do and many people don't want to know what I am currently doing. Many of my fellow researchers (Ph.D. students) are very good at organizing various events that happen in the department. I've also tried but have not been as successful. I mean to say: non-research activities also seem to be very important for a researcher and also as an academician.

To be frank, I don't like non-research events very much but still try to give my best. Many professors also know that I am not that good at organizing and managing events, but they always support me. I don't know how much this is going to hurt my career as a researcher and as an academician. I am an introverted kind of person and not that good at talking to other people (especially ones who I meet for the first time).

Question: How to grow in these ways as a researcher? After my Ph.D. I will be looking for postdoc or another job specific to research.
I would describe the areas you're concerned about as networking and service. 

While neither of these are critical path for your ability to succeed as a researcher, they most certainly will make you more well-rounded and can help you in many ways, so they are definitely worthy of investing a small but significant fraction of your time and effort.

What I would advise you, however, is that there are more ways to do networking and service than taking a leadership role in organizing events.  There are many other types of role that you can ask to take on that can ease you more slowly into such organizations and roles, but which won't push you out of your comfort zone too quickly.  Two good directions that I would recommend are:


Secondary event organizing positions, such as web chair, publication chair (i.e., the person who turns accepted papers into proceedings), local arrangements chair, tutorials chair, publicity chair, etc.
Reviewing for journals and conferences (often starting as a "sub-reviewer" for your advisor or other professors you work closely with)


The professors that support you can likely help you get such opportunities, and from these "training wheels" positions you can start to establish yourself as a known contributor in your professional community and start getting more comfortable with more visible roles and more active forms of networking.