The main influence that has been argued connecting Oppenheimer's interest in Eastern Philosophy and the atomic bomb is that of a sense of "duty." The Bhagavad Gita, which Oppenheimer famously quotes in his self-translated "I am become Death" line, is ultimately an expression of the idea of duty in war, with the line (and much of the epic) describing an un-reluctant prince being convinced to do battle by a god. In this reading, Oppenheimer — an intellectual with no prior interest in war — is the prince, the god is the bomb, World War II, what have you. I have written on this here at some length.

The other way to think about Oppenheimer's interest in Hinduism is as a matter of identity. Oppenheimer was born to wealthy Jewish parents on the upper-west side of Manhattan, and was confronted with anti-Semitism directly while an undergraduate at Harvard during the years in which its "Jewish quota" was most fiercely debated and defended by Harvard's leadership. It is clear that in Oppenheimer this manifested as a desire to associate himself with anything other than Jewishness, and this is the time in which he became very interested in trying on various other identities — such as the "rugged Western rancher" personality that later manifested as the choice of the Los Alamos site for the secret bomb laboratory. Oppenheimer's fascination with Hinduism — about as far as you can get from either religious Judaism or the Secular Culture (a sort of secular Judaism) his parents embraced, can be seen in this light as well. One can similarly see his interests in developing the bomb as being driven by identity issues as well: he had an almost pathological need to prove his loyalty and patriotism, one several authors suspect comes out of these initial identity questions. For a very nice dissection of Oppenheimer's psychology of identity and its manifestation in many of his activities, including his dogged determination to build the bomb, see Ray Monk's biography of him.

These kinds of answers are not mean to be monocausal — one doesn't say, "Oppenheimer made the bomb because he had been exposed to anti-Semitism as an undergraduate." People, and history, are more complex than that. But these influences do help us understand some of the myriad factors involved, and do help illuminate someone as complex and at times confusion as Oppenheimer.