This is a very niche skill. Each industry and department's need for Business Continuity will be different.
For example, Business continuity for a software development department could mean access to applications, tools, network etc. whereas Network Infrastructure department's Business Continuity could mean something entirely different. So what this means is what measures have been put in place to give alternate access to applications in case there is a network failure. As a Business Continuity Planner you will be involved in driving projects to enable this functionality to the business. Once its been developed, you may be responsible to execute it when needed.
In simple words, depending on the industy and department your job will be to make sure business keeps running no matter what.
Maybe its not the most straight forward - I think your first step should be deciding on which industry and/or department do you want to be a Business Continuity planner for.