I do this. It is called palatalization and is caused by the "tr" combo more than anything. The same process also occurs without "s", and with "dr". I often say, for example, "tree" as [tʃri] ("chree") and "drier" as [dʒrajɚ] ("jrier").
These variants don't come from Yiddish, German, or any other language. It is simply a natural phonological process that occurs in many languages, often due to co-articulatory factors.
So, in this particular case, it is caused because our English "r" sound ([ɹ] to be precise, but for simplicity I'll just use the symbol [r]) is articulated slightly further back in the mouth than [t] or [d]. In anticipation of that "r", the "t" gets pulled back, which sounds like [tʃ] ("ch").
When you add "s" into the mix, it just stretches this co-articulation further, causing palatalization of the "s" and the "t". You actually end up with something like [ʃtʃrit] for "street" ([strit]).
Why do people do it? Well, it is normal language change, but as it is a feature of certain dialects, it can make something you say come off as more dialectal; so, one would do this for the same reasons one would use other colloquial or dialectal language. I would be much more likely to say this around family and friends than at work; I consider it to be friendlier sounding when I speak that way to my wife, for example.
In some languages, a given palatalization process can remain a dialectal variant or only occur with free-variation between the standard and palatalized versions. But, it can also permanently establish itself in the language, and become a regular rule that is always followed.