It was part of the general Stalinist suspicion of Western and avant-garde influences. Jazz actually didn't fare too badly: it wasn't formally banned until the Late Stalinist (ie post-war period). For comparison, popular dances (eg tango, foxtrot and waltz) and Western popular music were all banned in the early 1930s as examples of capitalist decadence. Here too Stalinism meshed well with an older Bolshevik strain of puritanism. To quote from a Soviet factory newspaper in 1929:

The foxtrot was banned. Now in the ballroom, in public at youth parties, the foxtrot is danced under the guise of the waltz. When one watches the dancers, one sees what an aroused state they get into. It seems that we can expect nothing from such a waltz but depravity. That is why such waltzes should be forbidden at our parties. Parties are not for debauchery, but for the cultural rest of our youth.

As I said, jazz actually survived this first wave of cultural reactionarism and a genuine craze for it can be said to have swept the country in the 1930s (the so-called 'Red Jazz Age'). Jazz also nicely meshed with the new Stalinist love of 'cultural improvement' - an appreciation of jazz or the tango was a handy marker for a former-worker/peasant-turned-factory-director trying to impress his peers.

There were periodic attempts to discourage jazz, often on nationalist grounds, from the late 1930s onwards but it wasn't until the post-war period that jazz came under sustained assault from the state. Notable jazz artists were arrested and the sax was finally banned in 1949. The logic for this broader wave of cultural reaction - the Zhdanovshchina encompassed almost every medium - was a Cold War desire to combat Western (ie modernist) influences. Jazz was obvious expression of American culture in this context.

And that's as far as I'm qualified to take the story. I assume that jazz was 'rehabilitated' in the post-Stalin period (certainly notable artists like Leonid Utyosov were popular icons in the Khrushchev era and beyond) but that's outside of my area of knowledge.

Sources Mark Edele gives a quick overview of the fate of popular music in his Stalinist Society. Shelia Fitzpatrick's Everyday Stalinism touches on it as an example of Stalinist self-improvement. The foxtrot quote came from Kevin Murphy's Revolution and Counter-revolution.