How do you, as a SM, tell others in the company about what you are doing?

Why not offer them better than that? The Product Owner is accountable for value to stakeholders, potentially including others in the organization. He or she can invite them to a Sprint Review, where they can see the actual evidence of work completed.

He is afraid that people from outside the scrum teams may think that we're sitting and scrolling Facebook. 

Perhaps the right question to ask is "Why is there such a culture of mistrust in this organization?"

Perhaps the right question to ask is "Why is there such a culture of mistrust in this organization?"

I thought exactly the same at the beginning, after some time I realized that it was probably more about lack of knowledge. You more or less know what the recruiter or programmer does during the day. The role of scrum master is something completely new with us. The "classic" project manager has schedules, budgets, risk registers and other "serious documents" to show and Scrum master tells people that he makes it easier for the team to work. And while his team sees the work perfectly, the people from outside not necessarily. It can also be a great opportunity for education because more and more people in the company are interested in Scrum and in the end the role of SM is also to teach the organization how it works. 

Why is it needed for other people outside the Scrum Team that are not involved as either team member or stakeholder to know that the team is not just browsing around Facebook? 

Run a physical story board with 'To Do', 'in Progress', and 'Done' columns and move user stories on sticky notes accordingly (even if it's being done on JIRA or any other tool). I believe 'others' here might be senior management within the organization. One should be able to understand the status as soon as they see the physical board.

I can relate to your situation as it happened (happens) at my current company. Even the team members with which we work seem to think we spend our days on social media.  My fellow Scrum Masters and I have had this conversation MANY times.  What we have decided is that it isn't as much a mistrust of individuals as it is a mistrust of Agile/Scrum.  We are all working to make the work we do obvious.  We make sure to take every chance we get to work with people outside our immediate circle of influence to "evangelize".  A couple of the services to the organization stated in the Scrum Guide for the SM are

• Leading and coaching the organization in its Scrum adoption;
• Helping employees and stakeholders understand and enact Scrum and empirical product development;

We have decided to take those further and try to help others. We do presentations on Scrum to various groups that express an interest in learning. We offer to facilitate meetings for other groups.

As for a report for the organization we have never been able to come up with anything that shows what we really do. I am considering talking to our Human Resource Manager about whether are any kind of reporting historically done by them to show how teams are bonding and becoming more efficient. That is something that those organizations have done for a long time. 

Wish I had a better answer but in reality everything we do is only visible by the improvement of the team to work better in a self-organized, self-managed manner that improves their ability to consistently deliver incremental value.  But even that is subjective.