You should care if it matters to you whether your beliefs are correct.
To use a surveying analogy, it's like the difference between an open traverse and a closed traverse.
http://en.wikipedia.org/wiki/Traverse_(surveying)
In an open traverse, you start at a known point and establish new points in a line heading away from it. If your instruments and methods are good, you might be very precise... but you have no way of knowing how precise (or imprecise) you are.
OTOH, with a closed traverse, you start at a known point and finish at either another known point or your starting point. This lets you test how good you were, since all your measurements/inferences have given you a prediction that you can measure against some established standard.
Effectively, saying "why should I care if it's been established by anyone other than me?" is like saying "I don't care about finding out how accurate my beliefs really are."