I also wanted to give a very short answer.Let $p$ be a prime number. It is easy to see, that the binomial coefficient $\left\(p\atop n\right\)$ is divisible by $p$ for $1\le n\le p-1$. So the $p$-th line looks like $1,0,0,\ldots,0,1$ mod $p$. Then by the recursive definition of the Pascal triangle a new triangle starts at the left and at the right (until they meet in the mid somewhere). And this process goes on and on. Probably the line $\left\(p^2\atop *\right\)$ is also a line with this property, etc. This explains the recursive nature of this phenomenom.

I also wanted to give a very short answer. Let $p$ be a prime number. It is easy to see that the binomial coefficient $\left(p\atop n\right)$ is divisible by $p$ for $1\le n\le p-1$. So the $p$-th line looks like $1,0,0,\ldots,0,1$ mod $p$. Then by the recursive definition of the Pascal triangle a new triangle starts at the left and at the right (until they meet in the mid somewhere). And this process goes on and on. Probably the line $\left(p^2\atop *\right)$ is also a line with this property, etc. This explains the recursive nature of this phenomenon.