First of all you have to beat the Slides Technique Challenge to unlock the Super Slider game mode. All you have to do is complete the technique challenge, no score requirements have to be met.
Though it's a bit obvious from the Techniques Challenge, if you paid attention to that, the point of this game is to practice/learn slides. Doing slides, you control the blocks in this game. The main piece of advice is that you should use the pause trick. Have a controller handy, and pause the game so you can decide what your next move should be and get ready to do the slide.
Sometimes these blocks are a bit glitchy. If the note isn't sounding loud enough, you can either lose control of the block, or worse, it will start to freak out and bounce around all over the place. If it starts to freak out, pause the game immediately and try to slide it again. The best way to handle these slides is to do a quick, brisk strum, wait half a second for the block to start glowing (indicating that you have control of it) and then do the slide. Don't slide it too fast, just do it smoothly. As soon as you have the block where you want it, palm mute the strings immediately. If the block isn't quite where you want it, pause the game and adjust it a bit. If it's positioned correctly, you can let the block go down or open strum all the strings and make it go manually.
So this is like a basic block breaker game that you have to control with your slide notes. Put blocks of the same color together and they will burst and give you points. Chain reactions will give you more points, seemingly exponentially, with the more blocks you burst. One thing to note is that you should pay attention to the colors at all times. The colors always change when you go up a level, including the block you are about to play. Also, the yellow and orange are VERY similar in color, so make sure to pay attention to the shapes within the blocks when you have both of these colors on the screen.
You don't get any bigger multipliers in later levels, per se, but you will gradually get more and more blocks on the screen and therefore get bigger chain reactions. When you get to the mid forties in level, you can get a million points in a single move. Which is good, because when you get to level fifty, the blocks become too fast to control at all. In summary you will need a bit of luck to reach the fifteen million, but the main point is to be very careful where you put the blocks. Think out each move, because there are times when one mistake will screw you over and you will have to restart the game. Good luck!
The achievement will pop a few seconds after you obtain the required score, during the game.