It has to do with the percentage agreements made between Churchill and Stalin at the 1944 Moscow conference. Here is a literal paper form the said meeting on dividing countries after the war.

Churchhill came to Stalin during the war with the idea of a Sphere of influence for the USSR and the UK. Churchhill was afraid of the United States returning to its pre-WWII isolationist attitude leaving the UK alone in its dealings with the USSR and wanted to ensure that the post war Europe was favorable to British interests. His focus was on the Mediterranean sea, he wanted to ensure that Countries like Greece, Italy, and Yugoslavia, mainly so that shipping through the Suez canal, a main British supply route, would be undeterred by Communist interests.

So in dividing up the percentages and where the spheres of influence began and ended, most of Western and Southern Europe remained under the UK's influence, while Eastern Europe heavily remained under the USSR. The countries that were half and half were Yugoslavia and of course Germany but this was more under the occupational agreement between all of the allied nations, including France and the US.

Roosevelt's failing health forced him to have an ambassador go in his stead to this fourth Moscow meeting, however he was excluded in this private meeting between Churchill and Stalin. He reluctantly agreed to this idea of dividing up Europe, however he didn't agree with the idea of returning the Kings of Italy and Greece to power, instead wanting those countries to have republics with more liberal minded governments in charge. Roosevelt and Churchill often argued on the importance of Bulgaria, as Churchhill felt that troops should be deployed there as to not let the USSR occupy that region post war, but Roosevelt disagreed, preferring to focus on Northwest Europe. It wasn't until Field Marshal Alan Brooke, the chief of the Imperial General Staff, informed Churchill that the British army wouldn't have enough troops to effectively invade the Balkans and more specifically Bulgaria, that Churchill finally acquiesced allowing the USSR to maintain control of Bulgaria after the War.

When it comes to Poland, the USSR did allow a republic after the war to take over. However Stalin sent in troops to remove all anti communist parties before the election began. Poland really had it rough.

PART I

The 1944 Percentages Agreement (aka the "Naughty Agreement") has been mentioned, so let me provide some additional background.

It's worth noting that Soviet policy and postwar plans for Eastern Europe were not an unchanging, universal template, but a range of options. Admittedly, these all dealt with both facts on the ground (there being some 12 million Soviet troops in Europe in 1945) and Soviet needs in 1945 - the USSR had just defeated a devastating invasion that killed some 27 million people in Soviet territories, and Stalin and his associates feared the USSR being at risk of further invasions. So the idea of leaving Eastern European countries completely free to do whatever they like was not on the table.

One option was so-called "Finlandization". Finland had signed an armistice with the USSR in 1944 that saw Soviet troops enter the country, and required Finland to fight German troops there. The subsequent peace terms saw some Soviet troops stationed in Finland (notably at a naval base in Porkkala), the legalization of the Finnish Communist Party, and required that Finland essentially be a neutral country on friendly diplomatic terms with the USSR. This did involve Finnish politics self-censoring and avoiding open criticism of the USSR, but otherwise the Soviets left Finnish internal matters more or less alone. Finnish Communists had participated in the government in 1945, but by 1947 had been pushed out. Finland kept its multiparty parliamentary democracy and capitalist economy, and even received aid from the West, but turned down the Marshall Plan and kept its foreign policy nonantagonistic with the Soviets. Stalin is supposed to have regretted not occupying all of Finland, to which Vyacheslav Molotov replied "Ah, Finland. That is a peanut." The Soviet Union basically got what it needed, and had more pressing issues elsewhere.

The second option on the spectrum was an option that the Soviets tried in the 1945-1947 period, especially in countries like Bulgaria, Czechoslovakia, and Romania - large political fronts, coalitions of democrats, socialists, and members of local Communist parties, assumed office through elections, and attempted to gain popularity through reforms like agrarian reform (and by agrarian reform I mean the breakup of large landed estates and their distribution to landless peasants, not Soviet-style collectivization).

One reason the Soviets pursued this approach in the immediate postwar period was because of demands from the United States at the Yalta Conference (February 1945) and Potsdam Conference (July-August 1945) that free elections be held in Eastern European countries, specifically Poland, Romania and Bulgaria.

Hungary held such elections in November 1945, and the Communsts, much to their and the Soviets' surprise, lost - receiving only about 17% of the vote. Nevertheless, with Soviet troops still in the country (it was a defeated and occupied Axis power, like Germany), the winning non-communist party was forced to accept the Communists as coalition partners, and give them the powerful Ministry of Interior portfolio. Subsequent elections in 1947 saw the Communists improve their results a little, but their increased use of force and intimidation to split the non-communist parties and break off sympathetic factions meant that they more or less were able to strongarm a coalition government under their control, to push through nationalization and collectivization.

Poland was something of a fudge compromise - the Soviet-established Polish Committee of National Liberation (also known as the Lublin Committee) was to become the Provisional Government of Poland, instead of the Polish Government-in-Exile in London, but at Yalta Stalin promised that the exiled politicians could return and participate in elections. The Provisional Government was a coalition government with major positions controlled by the Communists, and by the time legislative elections were held in 1947, the independent opposition was so harrassed with arrests and suspicious murders, and the ballots so stuffed, that the pro-governmental Democratic Bloc won an overwhelming majority.

Great answer by TheHourOftheWolf! To add a little more; my understanding regarding Poland is that the Russians, immediately upon occupying the country, began executing and imprisoning anyone they felt might be a threat to the puppet state they intended to establish. Including members of the polish resistance who had fought the Germans. They did this in all the area's of eastern Europe which they eventually controlled.

Churchill was deeply concerned about the Russians and the fate of eastern Europe, especially Poland, but his army was spent and lacked the ability to oppose Stalin militarily. He thought Roosevelt naive in his dealings with Stalin, and there is some question whether his (Roosevelt's) declining health affected his ability to negotiate the final settlement of Europe near the end of the war.

Some in the US (Patton was one) saw the USSR clearly as a future enemy, and wanted to deal with them immediately, while our troops were mobilized. This would've been very difficult (short, perhaps, of using or threatening a nuclear strike) as the Russians occupied large amounts of territory, had 11 million people in uniform, and likely could've marched all the way to the English channel had they chose to. While the US had superior air power, it would've been herculean task with an unclear outcome to fight the Russians. The US public was interested in ending the war, not launching a second round.

In the end, the Anglo Allies conceded to what seemed inevitable at the time.

Another factor not mentioned thus far is the United Nations. FDR looked to it to maintain the future peace of the world, and his vision of it was of a great-power condominium in which the U.S., the Soviet Union, Britain and China -- the countries that he described as the "Four Policemen" -- would rule via the U.N. and knock everyone else's heads together if they looked like causing trouble. For that to work, the wartime alliance would need to be preserved into the postwar era. On the other hand, if that alliance fell apart, so would the U.N., and a Third World War seemed the likely outcome.

At that time, everybody was looking to a resurgence of Germany as the thing that would spark such a renewed global crisis. The U.S. didn't intend to maintain a large standing army to hold down Germany permanently; the British couldn't; and the Chinese were too weak and too far away to help police Europe. Only the Red Army could keep a renewal of German militarism and expansionism in check.

If obtaining all those benefits for future global peace and security meant turning an indulgent eye on whatever the Soviet Union was doing on its borderlands, well, no omelet is ever made without breaking a few eggs. And anyway, the only alternative to co-operating with the USSR was confronting her, and after years of the most devastating war in human history, that seemed like too ghastly a prospect to contemplate.