After digging for a while, I found the answer in a paper called Assimilation at a Distance by Wayne P. Lawrence.1 It's a little dated (2000), but also cites studies from 1993 and earlier describing this phenomenon.
The recent history of American English includes a sound change that seems to have gone unattested in the scholarly literature.1 This is the change of /s/ to /ʃ/ before /tr/ (i.e., a PHONEMIC CHANGE), which involves a palatalization of the initial sound in the cluster /str/, typically in initial position but not exclusively.
The author further mentions that "this phonemic change seems to be neither dialectal nor regional." However, it's definitely an American English peculiarity. It seems that it was first heard on television or radio broadcasts and became popular over these channels. Of course, the internet helps spreading such a linguistic "oddity" – and even I know I picked it up from television shows and continue to (subconciously) use this pronunciation.
Regarding the phonetic principles behind this change, Lawrence summarizes:
More concretely, this pronunciation is an implicit "assertion" on their part that in their grammar /r/, being [+ interrupted], is marked with respect to the opposition INTERRUPTED versus CONTINUOUS - exactly the same way that [ʃ], being [+ compact], is marked with respect to the opposition COMPACT versus DIFFUSE. The marked character of /r/ is prompted and confirmed phonetically, of course, by its high degree of retroflection in American English.7
This would explain the fact that we only hear this in American English. What's interesting is that the same does not appear to happen to "spr", which can be explained by the phonetic differences in "t" and "p":
I would surmise, therefore, that /str/ lends itself to realization as [ʃtr] while /spr/ remains unaltered also due to the phonological value of /t/ and /p/.
Another more recent paper was written by David Durian.2 It goes into the details of social background and "shtr" as well as providing some more explanations for its development.
1 – Lawrence, Wayne P. (2000) "Assimilation at a Distance," American Speech Vol. 75: Iss. 1: 82-87; doi:10.1215/00031283-75-1-82
2 – Durian, David (2007) "Getting [ʃ]tronger Every Day?: More on Urbanization and the Socio-geographic Diffusion of (str) in Columbus, OH," University of Pennsylvania Working Papers in Linguistics: Vol. 13: Iss. 2, Article 6