Without getting into formulas, I think the easiest way to visually explain what focal length is is to use an empty 35mm film slide as a framing guide. (Note, that as time goes on, fewer and fewer people know what a 35mm film slide looks like, so the visual guide is less apt...)
First, you have to explain that focal length is a property of the lens. Just like a milk jug might hold 1 or 1/2 gallon or 1 liter, or a certain water bottle might hold 1/2 liter, any particular lens has a particular focal length. (In this analogy, zoom lenses are like collapsible water bottles, that have a certain minimum volume when collapsed, and a maximum volume when expanded). Just as the volume is a property of that particular bottle, so is focal length a property of that particular lens.
(Note: I didn't have to use bottle volume for the analogy. I could have just as easily used the height of the bottle as the property. It doesn't matter — this is just an analogy)
Further extending the analogy, it doesn't matter if the bottle is full, or half full, or empty — the capacity of the bottle is fixed. Just so with a lens: it doesn't matter if it's focused far away, or up close — the focal length of the lens is unchanged.
Related: What is focal length and how does it affect my photos?
Now back to cameras. Lenses of different focal lengths change the field of view when mounted on a certain camera. Conversely, when mounting different cameras (with different film or sensor sizes) on a particular lens, the field of view is also affected.
Here's where the 35mm slide comes in when explaining to people: for a lens of a focal length ƒ (say, 50mm), if it were mounted on a 35mm film camera (the ones most people who used film cameras are familiar with), then you would get the same field of view as is you held a 35mm film slide at a distance of ƒ (50mm, or about 2 inches, in this case) in front of your eye.
Another example: early in the evening of a full moon night, when the moon is low on the horizon and it looks impressive, if you wanted to capture it at full glory, imagine holding an empty 35mm slide at arm's length (about 3 feet or so, or roughly 900mm) to frame the moon. When framed with a slide holder at that distance, the moon will fill about 1/3 of the height of the frame. So that gives you an idea of what angle of view a 900mm lens will have on a 35mm film camera (or a 35mm full frame DSLR).
Related: What focal length lens do I need for photographing the moon?
Now, if you're talking about a camera with a smaller sensor, such as a 1.5 or 1.6 APS-C crop sensor on modern entry- and mid-level DSLRs, then a 35mm film slide holder no longer works. The framing tool would have to be 1.5 times smaller. In this case, it would be 24 x 16 mm. Using the smaller "1.5 APS-C slider holder" as a framing guide, then you could put it at the lens's focal length ƒ from your eye to judge the field of view size.
Related: Does my crop sensor camera actually turn my lenses into a longer focal length?
This is the easiest way I have found to explain and visualize focal length, without diving into the maths with the thin lens formula and pinhole projection angle of view formula.