What is a Scrum Master?

The question, "What is a Scrum Master?" may sound easy to answer - RTFM, the Scrum Guide has a very concise section. But when looking at what a Scrum Master really does, it becomes a profound topic in and of itself.

The narrow definition

The most narrow definition is that it's the one person on a single Scrum team making sure the Scrum Guide is being followed and the team successfully delivers on their Sprint Plan. Most people work in organizations that think of this narrow definition and also behave likewise.

For people looking to move into such a role, the best starting point is reading the Scrum Guide, getting a few books (such as Geoff Watts: Scrum Mastery) and a certificate or two (PSM-I from scrum.org or CSM from Scrum Alliance come to mind) then start job hunting and learning on the job.
Sticking with the narrow definition is not a very fulfilling job, as systemic impediments never get addressed.

The broad definition

The most broad definition is that it's a person coaching an organization on becoming more agile, challenging existing paradigms and opening the way to more flexibility, and this means individuals, teams and managers on every level - from intern all the way to senior executive.

People

When it comes to being a Scrum Master in a broader sense, then Scrum is maybe 5% of the picture. In fact, Scrum itself will become less and less important and - just like the Agile Manifesto states - people and their interactions will become prevalent.

Agile Frameworks

The first thing you will need to explore is the other agile frameworks, starting with the team-level ones such as Extreme Programming and Kanban - and getting yourself familiar with the organization-level ones such as SAFe (Scaled Agile Framework), Nexus, LeSS (Large-Scale Scrum), DAD (Disciplined Agile) and S@S (Scrum at Scale). Of course, that doesn't mean you will be using all of these - but you should have a sufficient understanding to comprehend where they are useful and what their limitations are.

Management

Having a bag of agile theory under your belt, you need to move into organizational design and management by reading publications from authors like Drucker, Deming, Crosby, Taguchi, Taylor etc - and learning to compare and recognize their models.

Psychology

You will also want to explore the field of psychology to understand how and why people tick the way they do. That's a vast field, where you might want to reach into behaviouristics and conditioning (Pavlov, Skinner), emotions (Ekman), basic needs (Maslov), negotiation models (Ury), transaction models (Berne), habits (Duhigg) and many others.

Philosophy

To round this off, you may want to gain a deeper understanding of philosophy, which also has a lot to offer, with the most notable things you may want to get being logic (assumptions, fallacies, mental models), axiology and epistymology.

Economics

As the subject of how wealth works - i.e., how value is created, consumed and transferred, there's a strong correlation towards making a company successful: you have to learn how decisions are made, why agents interact the way they do, and what are the driving forces of an economy (both local and global.)

It's not very useful to do all of this upfront, but continuing to learn while you're on the job and working to continuously increase your sphere of influence within the organization.

Summary

The Scrum Master is a very deep role which, when taken with a deep, serious learning attitude, has a lot to offer and provides a tremendous field for growth. If you put your heart into it, you will reach tremendous awareness of and influence on your surroundings.
Don't let people tell you that a Scrum Master is "just a team role". Great Scrum Masters can make a massive impact on their organizations.