This is a version of the general phenomenon of fixed-segmental reduplication. Many languages have partial reduplication where a particular element is a fixed phoneme (e.g. di-dō from /dō/; gbi-gbona). A number (smaller) have complete reduplication: rimisirana-rimisirana. The English echo reduplications mentioned above combine those two properties; likewise, Kolami maasur-giisur ('men and the like') has a fixed constituent (the onset of the second syllable). Replacive vowel patterns are also found in a number of languages, such as the Munda language Gta', where root vowels are replaced in various ways for various effects (e.g. "gross", "tender", "different"), so fixed segmentalism is not limited to reduplication. (I would be remiss if I ignored Semitic root-and-pattern changes in vocalism).
The ultimate historical explanation for the specific vowel pattern found in English is not clear, but it is something that languages do. It is fairly common in "affective" word-formation (i.e. processes that convey an added nuance to a meaning, as opposed to being a grammatical inflection). There does not seem to be any extensive study that assembles the facts and tries to discern universal patterns (e.g. is [ɪ-æ] more common than [æ-ɪ]). If the English choices and orders of vowels is just one of a myriad set of attested patterns, then there isn't much to explain other than "what happened that this pattern got fixed and not [ɛ-u]?". It's really not know if there are significantly recurring patterns across languages.