The "simple staff" is a proposal for an alternative music notation. In particular, it is a prosposal to replace the 5-lines staff by a simple staff, in which the lines relate directly on the notes, accross all pitches. The goal is to make reading notes easier, especially for beginners, but possibly for experienced musicians as well.
In standard music notations, the notes are represented on a staff of 5 equally-spaced lines. The staff is drawn with a Clef, which sets the reference for notes : for example E is the note of the lower line of the Treble Clef.
The scale staff, instead, is drawn with a thick line on every C note and a thin line on every F.
I have just decided to move the thin line from G (previously) to F, for aesthetic purposes. The pictures and examples will be inconsistent until I fix all of this.
On a whole octave :
The lines can be also be drawn as high or low as needed to cover the range of notes in the score.
Compare the standard two-staff for the piano :
with the equivalent with the scale staff :
Certainly, for a beginner there is a simplification in the reading of notes : knowing the Treble Clef and Bass Clef is not needed.
Read the Wiki for more.