A shorter focal length producers a wider angle of view when we keep the recording medium the same size. If you draw lines to the edges of the sensor or film through the lens's optical center, this becomes immediately clear. For the same sized sensor, moving the lens further away (longer focal length) means that the triangle has much more acute angles, and moving it closer pulls them out to be more wide angle. I've illustrated this in the longer answer at What is "angle of view" in photography?, because "angle of view" or "field of view" is the term we use to describe this.
If, instead of keeping the sensor size constant you shrink it to match the decrease in focal length, you can keep the field of view constant. This is why zoom and cropping are effectively identical (although of course actual zoom with a larger sensor allows you to record more detail and more total light). It's also the source of the concept of "crop factor" — the idea that different focal lengths are equivalent on different formats (or conversely that the same focal length means something different depending on format). This is, unfortunately, the source of much confusion — see for example Does my crop sensor camera actually turn my lenses into a longer focal length?
And, it's also why the very tiny super-short-focal-length lenses on camera phones are only moderate wide angle — the sensor is correspondingly tiny. See So my iPhone 6 camera lens is as wide as my full frame 35mm DSLR lens? for more.
All of that doesn't stop us from using focal length as a shorthand way to reference field of view. Of course, even when shooting film there were always multiple sizes, large and medium format on the one hand or smaller systems like 110 on the other. But due to the gigantic mass-market dominance of 35mm film, we tend to use that as the reference point, and speak of "35mm equivalent focal length" (and for that matter, call the digital form of that format "full frame" even though other sizes are equally "full" for their design). You can see this in action at What is an ultra-wide lens?, where answers are largely framed using focal lengths on 35mm as the baseline.