Why and how did the Soviet Union extend its control over Eastern Europe?
The Soviet Union extended its control over Eastern Europe in the immediate aftermath of the Second World War. In one nation after another, Soviet troops and diplomats backed Communists, many of whom had fought in anti-Nazi partisan units, in the political struggles in the wake of the war. This process took different forms. In Poland, entirely occupied by Soviet troops, Stalin simply installed a communist government. The same pattern took place in East Germany and Hungary. In other nations, Czechoslovakia being one example, the process took longer, and happened through elections in which Soviet-backed candidates came to power and outlawed ohter parties.
In any case, by 1948, Winston Churchill's observation of two years earlier that an "iron curtain" of communism had descended across eastern Europe had come to pass from a Western perspective. From Stalin's perspective, establishing friendly governments on the western border of the USSR was the only way to avoid the kinds of invasions that Russians had suffered in repeated European wars. This was a defensive strategic move, not an aggressive expansionary project.