If a vector between two events $A$ and $B$ lies inside the light cone, then the time ordering of $A$ and $B$ cannot be changed by any boost. Thus, even though simulteneity is relative, we can still uphold a notion of causality in special relativity, owing to this order invariance. The time interval between $A$ and $B$ can of course change between observers, but the order cannot. If the vector lies outside the light cone, then one can always find a boost that reverse the time ordering of $A$ and $B$.
Given this wholly geometric fact of time order invariance for within-lightcone vectors, we can now make some physical observations and reasonable postulates. For example, suppose I cook boiled eggs for breakfast, and my fridge and kitchen is, say, ten light minutes away. Suppose I raise a flag (event $A$) when I decide to have boiled eggs for breakfast, and then event $B$ is my sitting down to eat them. Leaving aside the time needed to boil them (let's assume we have some futuristic cooking process that takes vanishingly small time), then if I could travel to my fridge at greater than the speed of light, some observers would see me eating my cooked eggs before I decided to eat them! Thus it seems reasonable, if we postulate that the orders of irreversible natural processes are not to be observer dependent, that travel at greater than $c$ is impossible.
Contrapositively, it is this geometric fact of the lightcone and the invariance of time direction of all vectors within it that even allows a notion of causality in special relativity, as long as we postulate that causally linked events can only be linked by within-lightcone vectors. It can be shown on very general symmetry grounds that the Lorentz transformation is linear and homogeneous and, if the transformation depends continuously on relative velocity, then this means that the only possible transformations mixing space and time are rotations or boosts. But rotations are ruled out if we want to uphold a notion of causality as discussed above, because one can always find a rotation to reverse the order of events. The lack of null vectors and light cones thus rules rotations out, so the transformation must be the signatured Lorentz transformation, with boosts rather than rotations.
This is what ACuriousMind, and the Wiki page, mean by the quote, "*[...]the light cone plays an essential role in defining the concept of causality" *".