Confusion understanding causality?
The statement is not correct. In a causal system, the cause (i.e. input) does indeed precede (come before in time) the corresponding effect (output).

The correct statement is the cause doesn't follow the corresponding effect.

"Cause creates the effect", so in a causal system, Cause ($x[n]$) will always come first and then the corresponding effect ($y[n]$). 


  A causal $h[n]$ can not use samples $x[n+1], x[n+2], x[n+3], ...etc$ in order to create $y[n]$. $h[n]$ can only use $x[n], x[n-1], x[n-2],...etc$.