Without going into the formulas, I think the easiest way to visually explain the focal length is to use an empty 35mm slide as a framing guide. (Note that over time, less and less people know what a 35mm film slide looks like, so the visual guide is less apt …)
First of all, you need to explain that focal length is a property of the lens. Just like a milk jug can hold 1 or 1/2 gallon or 1 liter, or a certain bottle of water can hold 1/2 liter, so any particular lens has a particular focal length. (In this analogy, zoom lenses are like collapsible water bottles, which have a certain minimum volume when folded down and a maximum volume when folded down). Just like volume is a property of this particular bottle, the focal length is therefore a property of this particular lens.
(Note: I didn't have to use a bottle volume for analogy. I could have used the height of the bottle as easily as the property. It doesn’t matter – it’s just an analogy)
Extending the analogy, it doesn't matter if the bottle is full, half full or empty – the capacity of the bottle is fixed. Just like with a lens: it doesn't matter if it's focused far or near – the focal distance of the lens is unchanged.
Related: What is the focal length and how does it affect my photos?
Now back to the cameras. Different focal lenses modify 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 vision is also affected.
Here is where the 35mm slide comes in when explaining to people: for a lens with a focal length ƒ (say, 50mm), if it was mounted on a film camera 35 mm (those that most people using film cameras know), then you will get the same field of vision just as you hold 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 want to capture it in full glory, imagine holding a empty 35 mm slide at arm's length (about 3 feet or or about 900 mm) to frame the moon. When framed with a slide holder at this distance, the moon will fill about 1/3 the height of the frame. So that gives you an idea of the viewing angle of a 900mm lens on a 35mm film camera (or a 35mm full frame DSLR).
Related: What Focal Length Lens Do I Need To Photograph The Moon?
Now if you are talking about a camera with a smaller sensor, such as a 1.5 or 1.6 APS-C crop sensor on entry level DSLRs and mid-range, a 35mm film slide holder no longer works. The framing tool should 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, you can place it at the focal distance of the lens ƒ from your eye to judge the size of the field of view .
Related: Does My Crop Sensor Camera Really Turn My Lenses Into A Longer Focal Length?
This is the simplest way I have found to explain and visualize the focal distance, without delving into math with the formula of the thin lens and the formula of the angle pinhole vision.