roujesky,
a picture is worth a 1000 words. Check out this link on BSP trees. <With regard to Darkbasic, most remarks to bsp are regarding a format for saving and loading information of a 3d world or 3d level.>
http://www.dgp.toronto.edu/people/JamesStewart/378notes/23bsp/
Some uses of bsp trees (simply stated):
helps with hidden surface removal. (not drawing faces of polygons that can't be seen from the players perspective/viewport).
helps to determine drawing order of polygons in a 3D scene.
Used to store information,location,geometry of a 3d world/object/level.