Quantifier statement explanation 
I'm having trouble understanding what the above statement means in plain English.
Does it mean: "For any Natural number there exist a number greater than it". I don't get what the right part of the implication means.

You are correct. Literally translated, the statement means "for all $x$ such that $x$ is a natural number, there exists a number $y$ such that $y$ is a natural number and $x$ is less than $y$".
