There is no doubt that Obama is media-savvy, so when he chooses a somewhat curious venue for a major speech, it is worth examining why. His speech at the New Economic School in Moscow on Tuesday is no different. A professor at that university speculates about Obama's reasons for choosing NES over more high-profile sites:
First, it is important that NES is a new institution, founded in 1992. Similar to the way Obama represented a new political paradigm for Americans who voted for him, NES also represents a new paradigm for the modern study of economics.
Second, Obama is the first U.S. president in many years to have built his political career not by dividing voters into separate ideological camps, but by searching for pragmatic and nonpartisan solutions. After 70 years of ideological domination by the state, the NES is continuing the traditions begun by such great Russian economists as Yevgeny Slutsky, Nikolai Kondratyev and Nobel Prize winner Leonid Kantorovich, whose work rose above ideological or national boundaries, as it should be in any scientific field.
More at the link, but maybe a more simple explanation is that Obama wasn't invited to a more prominent stage. After all, Obama has not shied away from grandiose settings for past speeches, and the Russians didn't seem interested in attracting a wide domestic audience for Obama.
Via Joshua Tucker at The Monkey Cage, who has more reactions to and context for Obama's trip to Russia.