Pretty much only attorneys and law students know what an LLM is. When I tell others that I am getting an LLM, they say, "What is an LLM?"
Anyone have advice on how to provide a brief and simple explanation to a layman as to just what an LLM is?

An additional law degree in the U.S., a Master of Laws

LLM is a degree available in most countries and hence it is famous and respected. However you can tell someone its a masters in the field of Law jus like MBA is a maters in business administration

Actually, I'm an American and getting an LLM to specialize in a field in which I am passionate about. When people ask what an LLM is, my responses have varied from:
1. It's an advanced law degree to specialize in an area of law.
2. It's a post-doctoral degree (although true, this seems misleading).
3. A Master's degree for lawyers.

I'm sure people out their can suggest a better response for when a layman asks, "What's an LLM?"

Your #1 answer sounds like the best. Maybe follow it up with number #2?

A specialization for lawyers, more or less like a MBA for others professionals. Just this.

If you say it is a post-doctoral degree, people will laugh at you in my country (you have to have 11+ years of Law School to claim this).

I know enough MBA holders (from top schools) who have told me that it is not a "specialized" degree, but a more general presentation of the various elements of business. So, I would not necessarily equate an LLM to an MBA. A JD is more akin to an MBA, as far as the level of focus on a particular area of subject matter. An LLM is nothing more than a traditional Masters (M.S. or M.A.) degree which, in essence, is a degree that introduces an advanced focus of a narrow subject matter.

Therefore, I would simply tell people that it is a "Masters Degree in (Insert Subject Matter) Law."

If you're a US educated lawyer, then say it's an advanced law degree to help you specialize in some subject matter.

Don't get into the whole Master of Laws description. Most people will be confused. If you're a US educated lawyer, then you just got a juris DOCTOR and now you're getting a masters degree. People will think you're taking a step back.

If you want to be really fancy, tell people you're going for a Legum Magister.

If you're not going for tax, and you want to be honest, tell people it's a way to avoid the job market for another year.

Last sentence ^^^ is most accurate.