Take a point $G$ on the elliptic curve $E$. Someone calculates point $P = h*G$ where $h$ is some secret number (this can be done with point addition and duplications fast). Your task: Given public points $P$ and $G$ and curve $E$, find the secret $h$.
Solving this problem is hard. That is the problem that makes elliptic curves secure.
Does a larger n value imply a higher security level?
Yes, the $n$ in your explanation is the order of a point on a curve, or put another way, it is the number of times you have to add the point to itself until you end up with the point at infinity $O$.
(There is also the order of the group of points on $E$ which is simply the number of points on the curve. Don't confuse these two orders, they are not the same thing).
Having a point $G$ with high point-order $n$ implies that the group has at least order $n$ as well.
So why do you want to pick a point with high order $n$?
The reason for this is, that if you pick a point with low order, there is a mathematical way that allows an attacker to solve the elliptic curves discrete logarithm problem faster. This is known as the MOV attack on elliptic curves:
Note: If you play around with curves (something I suggest you should do), you will find that some points can have a scary low order. I did some experiments with a toy curve in the past:
$y^2 = x^3 + 1001*x + 75$, modulo prime $p$ = 7919
This curve has group order 7888.
If you examine the points on this toy curve (you can brute-force everything) you'll find most of them have point-order 7888 as well. Point <4023, 6036> is for example one of them. The point with the lowest point-order I've found on this curve is <7285, 14> with order of just 6.
Due to the MOV condition the second point would be a very bad point choice as G on this curve.