It simply tells us how much energy we finish up with if we convert a known amount of stuff into pure energy: you'll probably never get around to doing this unless you become a nuclear physicist (and that's really difficult to spell so as a career choice, it's not for everyone).
Firstly, nobody ever talks about this, but we need to know what the letters mean and what the units are:
E is Energy measured in Joules
m is Mass measured in kilogrammes
c is the speed of light in metres per second (approximately 300,000,000 m/s).
So what's a 'Joule'?
One Joule is the amount of energy required to raise a mass of 1 Newton by 1 metre (at sea-level on Earth)
So if you could somehow convert 1kg of anything into pure energy you would release 1x300,000,000x300,000,000 Joules of energy
That's 90,000,000,000,000,000 Joules... a big number but what does it actually mean?
Well, the average apple has a mass of approx 1 Newton and there are approximately 10 Newtons in 1 kg so imagine you had a bag of 10 apples and you then lifted the bag up by 1m: as you've just lifted up a mass of 10 Newtons by 1m you've therefore used up approx 10 Joules of energy, so that gives you an idea of what 10 Joules-worth of energy feels like.
Now back to E=mc²: remember, converting 1kg of mass into pure energy would release 90,000,000,000,000,000 Joules which is 9,000,000,000,000,000 times more energy than you just expended to lift that bag up by 1m!
In words it's the energy you'd exerted to lift that bag of apples multiplied by 9 thousand trillion!
And that's why stars are so powerful because they weigh rather more than 1kg and converting their mass into energy is just about their favourite way of spending their time: they're not perfect at it (not by a long way actually) although they're much better at it than we are.
FOOTNOTE:
Theoretically (I think) E=mc² holds good for whatever the stuff is you're converting to pure energy, which for all I know might mean that a slab of chocolate cake could produce the same amount of energy as a slab of plutonium of the same mass (although a far better way to convert chocolate cake into energy is to just eat it).
In practice we can only really convert reactive stuff like plutonium and uranium into energy: the energy generated is released as heat which is used to boil water, produce steam and turn turbines to generate electricity. But even then, only a bit of the mass is converted to heat energy and what mass remains is nuclear waste. At the moment we don't know how to deal with nuclear waste as it remains radioactive for thousands of years: perhaps we should fire it into the Sun but instead we bury it underground for someone else to deal with in the distant future.