'Wear', as in 'wear out'. If something becomes 'worn out', its state has deteriorated due to constant 'wearing' or use. In this case, the carpet has become 'worn out' as a result of friction, or being stepped on. Perhaps an easier way to think about it is, a toothbrush. 'Wearing out' something always involves friction. Every time you brush your teeth, you wear it out a little, because of the friction. Over time, the toothbrush loses its shape and form, becoming frayed and soft. This is what happens when it is 'worn out'. Think of anything that you can wear, or use physically. They all eventually 'wear out'. The state of something being 'worn out' is what you see when you've used it a lot.

When something gets old it fades and thins out https://www.google.co.kr/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=to%20wear%20out%20meaning

When a carpet wears, it gets thinner, and if left too long, it gets holes in in. The soles of shoes also wear, when you wear them a lot. See - there are two meanings of 'wear'. One means to put something on, like clothes or shoes. Another is to get thinner, or loose some of its material.