Maybe an example would help explain. If we sketch this graph it is a bowel shaped object and cut along the horizon at some value of z = constant we get a level curve. I am right so far?
This curve is a circle that is 2 dimensional and sits in 3 D space at the height z?
And if I have all the curves then they form the set ? True.?
$f(x,y)=c$ gives you the circle of radius $\sqrt{c}$ in $\Bbb{R}^2$. This is the level set (in this case, curve) for the value $c$. Note that the general definition of the level set of a real valued function $f : \Bbb{R}^n \rightarrow \Bbb{R}$ at $c \in \Bbb{R}$ is $ \{ (x_1, \dots, x_n) \in \Bbb{R}^n \mid f(x_1, \dots, x_n)= c \} $. Note how in our case, the points of the level set will be living in $\Bbb{R}^2$.