There are actually quite a variety of reasons for why France was able to pick up its great power status after the war with relatively little fuss, and was treated as being an equal alongside the US, USSR, UK, and China.

To start with, being a great power and being an important member of the international community is more than just military power alone. Military power is a very important thing of course, and France was only marginal militarily by the end of the war, with her fleet sunk, an army that was just being reformed, and an air force that was of mediocre size. The USSR often stressed this lack of French material power in response to French responses for various positions post-war. However, France had certain advantages: for decades following the end of the war much of global diplomacy still was in majority in French (from my recollections at least - I don't have statistics about it to be honest), France had the second largest colonial empire, and France was a traditional great power, which meant that it maintaining that role was easily imaginable. Furthermore, France did continue to enjoy a great amount of prestige, especially in Latin America and Eastern Europe, and 5 was viewed as an ideal number of Security Council members. One of the great problems of the League of Nations was that it was never really universal, not the way that it was intended, so when the new attempt came, it was important to ensure the maximum amount of world legitimacy. There was a need to have states beyond just the big nations to ensure that it was viewed as a fair system for the smaller nations, and thus France and China served an important role in such regards, as spokespersons for smaller states. The same thing happens today - there is constant discussion about whether we need to reform the UN to better represent the needs of certain states and regions, and thus proposals for Security Council membership of states such as Brazil or India, to democratize and make the UN Security Council more representative.

In addition, while France was militarily weak, they were very, very careful to make sure that they were fighting everywhere, even if only in small numbers, so Free French troops were involved in all of the major theaters of the war : North Africa, Italy, various African colonies, France of course, Germany, even the Far East (where the French battleship Richelieu operated). Were the French contributions there always decisive? Sometimes they were of great aid - Bir Hakeim, the French expeditionary corps in Italy, French resistance forces in France - but regardless if they were or not, they were there, and De Gaulle's government was very good at utilizing its propaganda to show that France was fighting, and that it therefor deserved respect for military accomplishments as well, and to be able to counter the stain of Vichy.

Beyond this idealistic element, there were important negotiations behind the scenes. By far the most important, thing and the one which really led to all of the rest, was France getting a UN Security Council seat. This was not initially planned, and this depended upon a country quite distant from France: China. The United States under FDR supported China as one of the "four policemen" of the post-war era, one of a supposed 4 great powers who would be responsible for keeping the world safe and orderly (the others being of course, the US, the UK, and the USSR). Roosevelt felt extremely strongly on this matter. The UK and the USSR were... not so thrilled, as both viewed China as essentially an American puppet, dependent upon the United States and which would just vote the way that the US wanted. If the US was to get its wish and get China to become one of the big four (necessary to ensure the previously mentioned critical world legitimacy), then there needed to have some sort of counter-balance to the US. Who? Well, there's really only one other nation which might possibly be considered for this role in 1944: France. France was the proposed candidate of the United Kingdom, which viewed France's Security Council status as being a vital interest to provide a counterweight to the USSR and Germany, and to grant another colonial nation on the Security Council bloc - otherwise, there would only be Britain, while France would seek to defend the interests of the colonial system. Both the USSR, and the US, were still opposed to French membership. As late as April 1944, France's name was still not inscribed into the list of post-war Security Council nations announced by the US, and in July 1944, Roosevelt still was telling De Gaulle that there would be 4 policemen in the post-war era - and France was not one of them.

But things changed, and on July 18 it was decided by the State Department that France was included in the Security Council, as soon as it had an elected government. Britain's support, and the US desire to ensure maximum UN legitimacy, led to France being included in the Security Council, and the Soviets conceded before the Anglo-American position. Interestingly, Brazil was proposed at the same time, and in material factors wasn't too dissimilar, but both the UK and the USSR opposed Brazil's inclusion, seeing it as just like China : a way for the Americans to increase their influence through another pliable member. It is something of an irony that it was US support for China that played a critical role in giving France a Security Council membership, when post-war China became a Communist state which was opposed to the United States (for the first few decades at least), while France, initially opposed by the US, played a keystone role in the post-war US alliance bloc, but that's the way history works.

Once France had a Security Council membership, things fell into place much more easily, because now it had the authentic plumage of a great power and leveraged that accordingly. France managed to achieve membership in the European Advisory Commission, which was composed of the Big Three, the most important forum for European policy-post war. Initially only Britain supported France's candidacy, but the USSR and France had reached an alliance understanding at the time (the details of which I would have to track down), and so the USSR decided to support French membership too. The British were then concerned that the French would be drawn to the Soviet side, so encouraged the Americans to allow the French in, which succeeded. France received an occupation zone in Germany despite not being at the Yalta conference, which then meant that it was infeasible to keep the French out of the Control Commission for Germany. France was extremely displeased about not being involved at Yalta, and rejected any sort of alternative measure short of full-inclusion : it received just that, full-inclusion, at San Francisco when the UN itself was inaugurated. France was able to play itself as a great power and force enough sensitiveness towards that, that it could just pull off a thread-bare impression that it really was one. And at San Francisco, France (and China), received full rights as possessed by the Big Three in the UN, such as on the Council of Foreign Ministers.

So, that having been settled about how France was able to gain a status as a great power and receive an occupation zone, as a tl:dr about the why:

1)France was a traditional great power, so it was easy to accept it as becoming one again. It enjoyed significant prestige and non-military assets.

2)French participation in the UN Security Council was viewed as necessary to ensure acceptable international support for the UN.

3)Once France was a member of the UN Security Council, it had what passed for great power status in 1944, and it was able to capitalize upon this to steadily aggrandize other sectors where it was previously excluded from. Since it already had control over various sectors, if it was excluded from others then it could feasibly make life hard in those it had : thus when France was granted an occupation zone in Germany, then excluding them from the Control Commission was impossible.

4)France played whatever military accomplishments it had very skillfully, to generate least some prestige.

5)France was a vital geopolitical position for the emerging post-war order, ensuring its support by the United Kingdom as a buffer against both Germany and the USSR.

6)US support for China as a great power meant that the gate was opened for France to resume its great power status as well.

7)France could play both the USSR and the US/UK as needed : the UK supported it as a buffer against Germany and the USSR, while the USSR supported its membership in the European Advisory Commission as part of Franco-Soviet political understandings at the time, which then forced the US to support French membership.

8)French diplomatic leadership was very stubborn and very determined to ensure that France was treated as an equal : will alone is not enough, but the French were able to force enough sensitivity about their position to force their path back to great power status.

In contrast to France, nobody else in Europe had the same mixture of just enough material power to be credible, traditional legitimacy to be acceptable, and geographic positioning to be relevant, to achieve the same thing. France was just big enough and just important enough that they couldn't be treated like a small nation, and once that bridge was over, they were able to constantly capitalize on every concession to win another one, until they were accepted as an equal power alongside everyone else.

Most of my information here derives from What does it take to be a great power? The story of France joining the Big Five, by Gadi Heimann. There is also a book on French diplomatic history I read and which I recall some information on but it is a university library book and I don't have the name currently.