In what moment in the development of the German language were separable verbs introduced? Also, is there a linguistic reason behind their introduction? Thanks!
Separable verbs have actually not been "introduced" but actually have always been there so to speak.
Linguists posit that separable verbs are a primitive feature of Indo European languages and can even be found in non Indo-European languages (e.g. Hungarian).
You will find separable verbs in:
A. Modern Languages
B. Ancient Languages
First there is the familiar Latin sentence structure where, as in German, infinitive verbs are thrown at the end of the sentence in LIFO order. Did Latin ever have separable verbs? As a matter of fact "Old Latin" and a few other ancient languages did. As I read in this article of which I reproduce here the relevant part.
In Hittite at least the technical term is preverbs. Except for two of them, they are all separable. The same (directional) words can be used either as adverbs, prepositions or preverbs: anda (into), appa (away), arha (outward), katta (downward), para (forward), sara (upward). Looks familiar? Google "up" for "Hittite preverbs" for more.
Maybe the right question then could be "When and how did prepositions become [inseparable] parts of verbs?".
1 At that point, it is worth noting that Hittite is recognised as one of the oldest known Indo-European languages, if not the oldest — with such primitive characteristics as only two genders (animate/inanimate).