I have been reading about wild knots in $\mathbb{R^3}$ that have nonsimply connected complements. I'm a bit confused here and I added a picture of a wild curve with two circles in its complement. Is it so that the blue circle can be shrink into a point within the complement, but the red one can't be? On the otherhand, the final blue point lies on the wild knot itself? I hope someone can clarify to me that is there a connection with the properties of complement and the nature of wildness?
Both (blue and red) curves represent non-trivial elements in the fundamental group of the complement. That is, none of them can be shrunk to a point.
To see this, consider the linking number of these curves with the knot itself: it's $2$ for the red one and $1$ for the blue one (assuming consistent orientations). Since the linking number is invariant under deformations, it has to be zero for nullhomotopic curves.
This being said, your initial statement is not what makes wild knots weird compared to tame knots: indeed, the complement of any knot is not simply connected (since a curve such as the blue one here will always have linking number $\pm 1$ with the knot; it's called a meridian curve). What you probably intended to say is that