Although I haven't heard of the term "degrees of passive/active" before, they are almost certainly talking about the verbal stems. This is a concept indeed alien to Western European (or broader) but common to all Semitic languages. The core idea is that the stems differentiate voice and Aktionsart.
In an earlier stage of the language (pre-1000 BCE) there must have been ten verbal stems, nine of which fit into a 3x3 grid of voice and Aktionsart:
Unmarked Aktionsart Verbal plurality Causative
Active G D C (Š)
Reflexive Gt Dt / tD Ct (Št)
G is called G for the German Grundstamme. D is called D because of Doubling of the middle radical (/qatal/ > /qittel/). C is called C for Causativity. It is also called Š sometimes which better reflects phonology (although in Hebrew the marking consonant became /h/). Dt is often called tD because the /t/ became a prefix rather than an infix elsewhere.
Verbal plurality can be of subject, object or action, e.g. "they broke the glass" in G becomes "they shattered the glass (in many pieces) in D via object plurality, or "they repeatedly broke the glass" (action plurality) or "they all broke the glass" (subject plurality). Causativity is e.g. to cause to be seen, i.e. "to reveal". Thus you see that these distinctions are in English covered by lexicon rather than grammar (i.e., there are different verbs for different Aktionsarts, "broke"-"shattered" and "see"-"reveal").
Besides this nice symmetric system there was an N-stem, the original semantics of which are still debated but which probably had to do something with middle voice.
In Central Semitic, "internal passives" are developed for the three core stems G, D and C. In Arabic grammar these are usually not treated as separate stems, but in Hebrew and other languages they are:
Passive Gp Dp Cp
This system developed differently in different languages. I cannot comment on the situation in Arabic (which as far as I can see from Wikipedia must also have developed the new L stem) because I don't know enough about it. In (Biblical) Hebrew, there was the issue that Gt, Gp and N both indicate relatively rare non-active voice unmarked for Aktionsart. This leads to the dropping of Gt and to a large extent that of Gp as well, whereas N takes on passive meanings and Dt takes over reflexive voice unmarked for Aktionsart. Also, Ct is almost not attested in Biblical Hebrew and as far as I know does not exist in Modern Hebrew because reflexive causativity is so rare. Thus in Hebrew the situation has become far less symmetric than it once was. On the other hand, in Syriac, the function of the passives was taken over by participles and the N disappeared, leading to a highly symmetric system of only 6 stems (in fact, like the original table above).
After this long detour I would still like to comment on the terminology of "degrees of passive/active". I'm sure that the author had his reasons for this term, but it doesn't seem to be accurate: it covers only one dimension of the system (voice) but neglects the other (Aktionsart). Thus, the verbal stems G, D and C all indicate active voice (the same "degree of passive/active"), whereas Gt/Dt/Ct all indicate reflexivity and Gp/Dp/Cp all indicate passive voice. (Only in Hebrew the system is more complicated, because N can indicate both middle and passive voice, thus being in between degrees).
Lastly, for your convenience a table of the more common names of the stems in language-specific grammars:
Heb. Arab. Heb. Arab. Heb. Arab.
G qal I D piel II C hiphil IV
Gp qal-passive – Dp pual – Cp hophal –
Gt – VIII tD hitpael V Ct – X
N niphal VII
A good diachronic overview paper with many references is Gzella, H. 2009. 'Voice in Classical Hebrew against Its Semitic Background', Orientalia 78(3), pp. 292–325.