"In case they are proved right, what would the implications be?"
It's hard to say because we don't know if it's true and then what theory will fit that observation. And also, I'm not an expert on Particle Physics but I'll still give it a try...
Einstein's theory of relativity has been proven right very very often (and also it has not been proven wrong). The assumption that the speed of light is the fastest that any particle could travel is pretty solid. If it is proven wrong it's like knowing that atoms are made of smaller particles like protons, neutrons, electrons.
The speed of light is something that strong that even the researchers who found those fast neutrinos don't actually believe their results... Of course within a couple days, there has been published a counter paper stating that the researchers didn't read out the clocks right.
Experimental Physics is all about reproducing results to confirm if it's valid, especially if several experiments with different setups all over the world do. So this is also what the group aimed for but it is very difficult to detect neutrinos because they hardly interact with matter. As I read, there are only two other research facilities who could manage that: the Fermilab in the US and another group in Japan but the Japanese still suffer from the earthquake and Fokushima so they will need even more time. And another point is that neutrinos in other parts of the world are very hard to find. The CERN produces a lot of neutrinos, therefore the first results came fast but it will take a long time to have sufficient statistics to confirm or defeat the superluminal neutrino velocity (isn't that a great name?).