Until then the usual idea was that light must travel in some medium, the aether - and then - their idea was to find out how fast the Earth is traveling through the aether by measuring the speed of light in two directions at right angles and comparing them.
It would be a bit like a boat sailing in the sea - and creating waves - and other boats creating waves - and they go at the typical speeds of waves in the water - so you'd expect that if you travel fast enough in a boat - you might be able to catch up with the waves, even overtake them.
But - surprisingly - the experiment turned up a total negative. No matter what direction the Earth was moving - and no matter how they turned the experiment around - they always found no change at all in the measured speed of the light.
It's like - you see someone cycling away from you and they get a mile away from you every five minutes say - and you try running after them - but no matter how fast you run - they still get a mile further ahead of you every five minutes. Even if you run in the opposite direction, away from them, then still, looking back and measuring - you still find they are getting exactly a mile further away from you every five minutes exactly,
But someone else, standing by the side of the road perfectly still, while you were doing all your running - they measured them also - and again - found they are going away from them also, at exactly a mile every five minutes.
If you chase light, or run in the opposite direction, away from it, then it makes no difference at all to how fast it recedes from you.
It doesn't seem to make sense does it?
But that is how light works. That's what the experimenters measure.
And - that's been confirmed in all experiments since then.
Einstein then turned that on its head and said - suppose we just take that as our fundamental principle - that light is always measured to be the same speed relative to you - no matter how fast you might rush in your rocket in an attempt to catch up with it - or run in the opposite direction - it never seems to be moving away from you any faster or slower.
That's basically the principle of Special Relativity in a nutshell.
A bold and radical and - in a way rather crazy idea :).
Quite paradoxical seeming at first - not like the way we expect speeds to work. But - when you work it out in detail - bearing in mind - this is the key which makes it all work - that we need to use light itself to do our measurements of speeds and timing - it turns out to be all logically and scientifically self consistent.
So - all our experiments back that up. So - in a way - you can't go close to the speed of light. However much you try to catch up - it still goes away from you at the same speed as before.
To others it might seem that you are traveling at close to the speed of light.
To you it might seem that you are stationary relative to the moving light - and they are going at close to the speed of light in the opposite direction.
But - there is no right answer there. In Special Relativity at least - there is no "preferred rest frame".
None of you is really going any faster relative to light, at all. Just relative to each other but not relative to the speed of light.
Even if I go at 99.99% of the speed of light - at least so it seems to you - I will see light traveling away from me in all directions at exactly the same speed as before - as will you of course - and I will see you going away from me at 99.99% of the speed of light.
And - another funny thing - time slows down so that if I am traveling at nearly the speed of light - then to you, as it turns out - my clocks seem to be turning around far slower than yours (e.g. perhaps my clocks seem to be advancing just one hour every day of your time) - but if I look back at you, your clocks seem to be turning around far slower than mine (to me, yours seem to advance just one hour for every day of my time). So we both see each other apparently living and growing and doing our things at an extremely slow pace.
As for light itself, then according to this theory - no time elapses at all for the light. Light that left the Andromeda galaxy - as we would think - a couple of million years ago - spends no time at all on its journey to our eyes - in its own frame of reference. It arrives here "in the same instant" that it left the distant Andromeda galaxy.
There is also a strange effect on distances - that we measure distances differently from each other also.
You need all those things to be true for the special theory of relativity to hang together and be consistent - these effects of speed, time, measurement and distance.
When you do this, and work out all the details, then as Einstein found out - you get a consistent, and useful - but rather puzzling - scientific theory. It works and has passed all its tests so far - and that's our Special Relativity :).
If you find it hard to get your mind around that and make sense of it - then that's probably a good sign and means you have "got it" :).