Well Mr. Garrison was pretty ambiguous regarding a lot of things if you ask me, and Oliver Stone took much of that and ran with it. Here's a little more background...
Oswald was active in Socialist/Marxism since at least the age of fifteen. In 1956 he wrote the following...
Dear Sirs;
I am sixteen years of age and would like more information about your youth League, I would like to know if there is a branch in my area, how to join, ect. I am a Marxist, and have been studying socialist principles for well over fifteen months. I am very interested in your Y.P.S.L.
Sincerely, Lee Oswald
In October of 1959 having already entered the Marines Lee visited the American embassy and asked to defect to the USSR, apparently annoyed by American socialists as nothing but armchair socialists. He lived in Minsk for a while and eventually met and married Marina. By early 1962 he decided to return to the States, noting the decrepit conditions, poor pay (working at an electronics factory) and lack of entertainment.
His experience in Russia seems to have changed his views somewhat as he was also quoted as saying, after having lived both in the USA and USSR, "I have lived in both systems, and I despise the representatives of both systems"
Upon his return he of course settled in Dallas. The White Russians as the FA pointed out, were an anti-communist faction of Russian American emigres. In April, 1963 Oswald became enraged with former Major-General Edwin Walker and actually fired a shot at him in his home. He missed and later escaped undetected. So Kennedy was not the first person he took a shot at in his life with the intention of assassinating.
His marriage with Marina was on the rocks at this point, but having lost his job in Dallas she convinced him to head to New Orleans and return to his hometown. Oswald then forms the pro-Castro "Fair Play for Cuba Committee" and his views get him on a local radio program which was highlighted in Stone's film. Marina later moved back to Dallas to stay with a friend, and Oswald, still active in NOLA, rented an apartment room where he could visit his family on the weekends...his relationship with Marina still troubled.