For(F,L6+12,L6+767       //loop through all of the screen pixels except the last 12
{F} and {rand^8+Str1}->{F-12}   //take the byte, use AND to erase a single bit from it, and then store it into the byte directly *above* it
End                     //this makes it so that as the byte rises, each frame a pixel is erased from it

Thie first line loops through all the bytes in L6 (except for the last row, which is the last 12 bytes. We save those so that there's something to "feed" the flames). Since L6 has 768 bytes, we loop from 12 to 767.

If you look at the Str1 sprite, it's essentially diagonal pixels filling up the sprite, and each row/column has exactly one pixel. By using rand^8, we select one row at random, and erase that pixel from the screen. We then shift the entire screen up a row. By adding more pixels or taking them away from the sprite, you can do all sorts of things with the flames - bias them to one direction, make them last much shorter times, etc.

It does not work when I do it...

When I compile, an error occurs:

EDIT: It stopped.  Thank you, though...   

Also, why do you only need 768 bytes for 6144 pixels?

Each byte is made up of 8 bits. Since the display is purely monochrome and doesn't support grayscale, you can store one pixel in each bit, and 768*8=6144.