Or which phenomenon is causing this? Is there a known reason or rule behind this?
This happens with all Greek words in r- when a prefix is added, not just that particular word.
A likely explanation (though I don't have a reference for this) is that the sound written as Greek initial ῥ- was a long trill [rr], just like initial r- in Spanish. (The fact that ῥ- always comes from a cluster, *sr- or *wr-, makes this plausible.) Therefore when a prefix was added you need a way to distinguish this in spelling from single [r], leading to the spelling -ρρ- (or -ῤῥ- as it's usually written).
I would have to go look it up to be sure (or you could yourself in Andrew Sihler's comparative grammar of Greek), but I suspect that it has nothing to do with any influence from the dia- prefix, but is a remnant from the much older way of pronouncing any word that began with rho, sc. with an initial w-.
The initial w- disappeared throughout, but retained some traces in writing, including for an initial rho by giving it a spiritus asper. I suspect that not being able to write this spiritus in the middle of a word may have prompted the writing of a second rho.
The second rho is therefore not an assimilation, but a way of notation for an old sound that only left superficial traces in writing.
[Do feel free to edit my answer; I am writing this while very tired.]
Well, in the end it does not actually occur with all instances. A diphthong is not "allowed" to occur before -ρρ-, which means that words with the prefix εὐ- + ρ will not cause gemination, e.g. εὔρους, εὔρυθμος, εὔριζος etc. But, if you split this particular prefix to its possible incontracted form, the phaenomenon will occur, i.e. ἐύρρους, ἐύρρυθμος, ἐύρριζος etc.
We can claim, then - more generally and practically - that whenever one vowel meets the initial ῥ-, it causes a -ρρ- to "save" somehow the presence of h, as also suggested before. This includes compounding, too, as in Greek grammar the concept of "composita" includes not only prefixes, but also whole words (just to disambiguate from today's definitions of prefixes and compounds). Examples: καλλιρρήμων, ὑποδηματορράφος, φυλλορροεῖν, πτερορρυεῖν etc.
I thing TKR's and Aorists' answers combined hold the key; the s developped into an h-sound (like most initial sigmas) which caused the subsequent rho to be aspirated and voiceless (al aspirated consonants in Ancient Greek were voiceless contrary to Sanskrit which has voiced aspirates as well).
The result probably was an initial rho that took a slightly longer while to be pronounced than a single rho; in most compounds with words that have initial rho this rho is doubled to mark the geminate (and voiced?) pronunciation after prefixes and prepositions.