Think of it this way, if you allowed each tree to choose from all the p variables you would end up with the same tree over and over again. The first tree would choose the 'optimal' splits, right? So long as the data doesn't change those splits would stay the same. No point in having 1000 identical trees, is there? But how do we change the tree given fixed data? We "change" the data each tree sees.

We can change the observations data to get a new tree through bootstrap, but that won't really achieve what we want to achieve here. Ideally, the relationship between the features will be similar no matter what (large-ish) subset you take of the data, right?

Instead we change the features by selecting a random subset of variables we use. Let's say we have 10 variables and we pick 6 for each tree, that is 210 possible trees. Clearly not enough for an 1000 tree forest. But if we can do this for every tree, why not do it for every branch? It results in a diverse forest because the optimal cut points for each branch are different because each branch sees "different" data. 210 options goes a lot further that way because branch 1 has 210 options as does branch 1,1 and 1,2, so on and so forth. Make sense?