Is there a specific reason for which diphthongs in German and English words like "mein" and "like" arose? It seems to be a pretty common phenomenon, but somehow it seems to be limited to Germanic languages. The same question could apply to "Haus" and "House" etc.
The kind of diphthongization found in Germanic as exemplified by "like" and "mein" (or the reverse order of elements, as in a rising diphthong) is extremely common in Germanic, but not limited to Germanic. It also arises in the history of Spanish and Italian (dialects), Old French and Quebec French; Finnish (dialectally) and the Saami languages; Sakha (Turkic); Slovak; Guerze; Kɔnni; Yokuts, and Lama. Except in the puzzling case of Lama and possible some other Gur languages where only short vowels diphthongize, it happens to long vowels.
We typically notice these processes when they are firmly established some hundreds of years after the diachronic tendencies have started, but I think it is possible for linguists to document these tendencies now and see how they work out in a few centuries. For example, from the nature of the North Saami diphthongizations, comparison with corresponding processes in other Saami languages, and an investigation of the realization of the excrescent vowel attested in "extra-long" consonant clusters (bár(ə)tni), a good case can be made that [uo, ie, ea, oa] are a development from [uə iə eə oə], earlier [u: i: e: o:]. There is is a similar process ongoing in Norwegian where [e] is not pronounced as a monophthong, it is a phonetic diphthong somewhere in the range of [ɛa] or [ɛə]. Although it is known that American English [ɪ,ʊ,ɛ,ɔ] are usually realized as phonetic diphthongs, this is a low-level detail of pronunciation that doesn't yet enter into the phonology, so it is often ignored.
It is likely that this is a recurring instance of "gesture alignment", where the articulatory gesture characterizing a kind of sound is aligned more to the left, or to the right, within its domain. This is quite well known in the case of pitch, where the most common trend is displacement to the right. Applied to a vowel, if the articulatory gestures characterizing [i] are more timed towards the beginning of the syllable, you will tend to get an [iə]-like output and if they are more timed towards the end, you will get an [əi]-like output. The correlation with long vowels is explained by the fact that they have longer duration and therefore more time within which a perceptible distinct vocalic articulation can be realized. I do not know why Bantu languages so resolutely avoid long-vowel diphthongization.
True. The other long vowels were raised but /iː/ and /uː/ can't be raised further. The diphthongization was surely partly a strategy to differentiate the raised /eː : oː/ from /iː : uː/ (chain shifts/ push/pull chain).
Why these whole significant vowel shifts happened though is a good question. It seems that vowels in Germanic languages are very unstable, which leads to the Germanic languages having very many vowel qualities. This may be related to the Germanic word stress.
Diphthongs are not limited to Germanic languages. It is pretty common in the languages of the world.
I can cite four phonetic aspects that lead to the outbreak of diphthongs. First of all, there are the diphthongs stemming from long vowels. This diphthongization is found in English (see: https://www.ling.upenn.edu/~kroch/courses/lx310/handouts/handouts-09/ringe/gvs-revised.pdf)
The other origin comes from the fall of a consonant. This lost consonant has allowed two vowels to merge together and if they have different qualities, they became diphthongs. This consonant can also be vocalized bringing about the same thing. Such a evolution can be found in Riffian (see: http://www.academia.edu/29199244/Tarifiyt_Long_Vowels_and_Diphthongs_Independent_Phonemes_or_Simple_Phonetic_Variants_of_the_Basic_Amazighe_Vowels_2004_)
The third case is when a vowel is stressed, a breaking vowel can occur. This process existed in old French, the stressed Latin vowels had been broken (see: https://halshs.archives-ouvertes.fr/halshs-00365006/PDF/vaissiere_1996_from_latin_to_modern_French.pdf)
The last phenomenon corresponds to the assimilation or coalescence. When two sounds are adjacent, in this case a vowel next to a consonant or vowel, and where one takes a feature from the other, that leads to a diphthongization. Hausa is a language where this kind of assimilation have been observed (see: https://scholarworks.iu.edu/dspace/bitstream/handle/2022/21060/HausaDiphthongs.pdf?sequence=1)