Note: I'm not an expert in ancient notation (and English is not my native language), so please forgive me for using wrong terms.
What we're seeing here is not "really" a staff notation, but actually an ancestor that shared common aspects with tablature.
Finger/fretted instruments used lines to indicate each string, and numbers placed on each line for the finger or fret to be used for that string.
Keyboard instruments used a similar pattern, but since keyboards obviously don't have strings (and have lots of keys), the pattern was more similar to that of vocal tablature, with both lines and spaces used to place notes.
The reference for those lines and the resulting pitches depended on the instrument, and one might assume that what became standard staff notation is due to convenience aspects: using a standard notation system makes it easy to play/sing the same piece with different instruments, even if the player doesn't know the technical aspects (string intervals, fret spacings, keys) of the instrument for which that piece has been written or transcribed. Vocal influence (which used notation based on pitch) was also an important aspect.
At that time, there was no absolute standard for "staff notation", and staves with different line count were used even in early 1600.
The image you're linking is very interesting, as it shows a passage of that moment in music history for which there was no common standard yet (while already using some aspects of notation we still use nowadays):
- clefs are already clearly placed (the bass key indicates the F, the treble the G);
- there are alterations indicating the tonality (interestingly enough, it's using those of C-dorian, instead of C-aeolian);
- quavers and semiquavers are "empty" when the meter is based on the minimum (3/2);
An (un)educated guess
The reason for the line count and reference is unknown to me, but I'll give it a guess anyway.
As already said, I'm no expert in the field, so I'm writing the following only based on personal research and experience: it's an assumption, it makes sense to me, but I'll be glad to be wrong. Please comment or write your own answer if you do know more on the matter, and I'll be happy to rectify the following.
I believe we should consider that, since notation was not a strict standard yet, composers choosed their own reference for staves, based on the instrument they were writing for.
Harpsichords usually have a 4½-5 octave range, often starting from the F ("F2") two octaves below the "middle-C" ("C4"), which gives us the base common lower note as that under the first line on the bottom staff, but yet with some models going to the C below that (allowing the lower notes in the given score); the "added" lines above are probably to provide a better visual reference for intervals used within the piece, instead of constantly using ledger lines.
The upper staff is based on the middle C (which makes sense, as C was already considered as a reference), and extends to a reasonable height considering the range of the piece and the basic range of the instrument, in the case of a "standard" keyboard extending to F6.
Notes and references:
·  See the Staff article on Wikipedia, which supports the same based on the Harvard Dictionary of Music.
·  Almost any recording we can find seems to be played in B minor, but remember that the usage of A at ~440Hz is much more modern, and almost any musician playing ancient music uses a much lower tuning (~415Hz) which often results in listening to about a semitone lower than what's being written;
·  using the scientific notation that gives us the middle C as
C4; note that in electronic instruments is not uncommon to have a
C5 for the "middle C" (MIDI note 60);
·  we are now used to music "played as it's written" (or recorded, a concept that obviously didn't exist at the time); remember that notation was never considered to be the absolute depiction of what "should" be played: as much as a written theatrical play, notation is more like [one of] the beginnings of the music, not its performed result; early written music was more of a guidance than it then became in later centuries - and, interestingly enough, more similar to what became again with the new contemporary music starting from early 20th century (see Aleatoric music and including jazz/pop music;
· About clavichord octave range (and other aspects): https://www.saladelcembalo.org/instruments/myths30.htm
· Staff notation and line count in early keyboard music: https://www.britannica.com/art/musical-notation/Evolution-of-Western-staff-notation