gfugal, I hear you, that saying confused me for a long time too, still does sometimes. At the very real risk of making it more confusing for somebody, I'll offer my current take on it.
If more words were used to make it more specific (albeit less pithy), I think the saying would be:
Thickness determines the minimum radius to which the wood can be bent without taking set or breaking; for a given thickness, width determines how far the wood will bend when subjected to a given draw force on the string.
If I have a segment of limb of a certain thickness, I can bend that segment to a very specific minimum bend radius without damaging the wood.
If I have a segment of limb of a given width, it will bend when the string is pulled. If I make that same limb segment narrower and pull on the string with the same force, the limb will bend farther.
The practical implications being, among others: for a given bow length and draw length, the best bow is one that is everywhere thinned only as much as it needs to be to allow the necessary bend, while also being only as wide as it needs to be to achieve the desired draw weight. Which of course is the basis of the mass principle, etc. etc.