Explanations

The lyrics are not really important in this song (in my opinion). The singer just wants to transmit us an idea of freedom, so when you’re dancing in the disco and you hear things like "rise up", "over the rainbow", "direction sky", "I broke my chains", etc., you easily feel lighter, floating, and happy. That’s the idea. It’s a feeling, more than a meaning.

RISE UP= Get up.

DON’T FALLING DOWN AGAIN= Well, this is certainly not English grammar. There are two possibilities here:
1- It’s some kind of substandard black English, maybe Patois or something. The standard equivalent would be "don’t fall down again" or "not falling down again".
2- It is "don’t fall in down again". Still strange, but correct grammar. To "fall in" is to collapse inwards, so the "down" particle here could be emphasising the idea of crumbling (falling), because if something falls, obviously, it falls down.

LONG TIME I BROKE MY CHAINS= I broke my chains a long time ago (now I’m free, liberated).

I TRY TO FLY A WHILE= I try to fly for a moment.

DIRECTION SKY= To the sky, towards the sky, in that direction.

RAINBOW= Yeah, I know that you know what a rainbow is, but here are some pictures anyway: rainbows.

RISE UP WILL BE THE GAME... FOR MY MIND AND MY BRAIN= I intend to devote my energies to rising up.