De Gaulle resigned from his position in 1946 and stayed out of power for a very long time, until 1958. This period is usually called ‘La Traversée du Désert’, the ‘Crossing of the Desert’. It is not entirely clear why he resigned, but the most usual interpretation is that he refused to become an average politician in what he considered to be a mediocre regime. The constitution which had been drafted to replace the old 3rd Republic with a new Republic (called the 4th) had created a parliamentary regime with very limited powers for the President of the Republic. In particular, the President could not dissolve the assembly. Most of the time French politics were stuck in deadlocks between warring parties in the Assembly. The 4th Republic was very unstable, with governments usually lasting for a few months before the inherent instability of the system as well as the external circumstances (economic crises, failure of the Suez expedition, war in Algeria) would bring down the government.

De Gaulle despised this regime, and he wished for a strong and stable government. In his youth he had been an admirer of Maurras, a monarchist, and his level of commitment towards democracy was frail. His absolute ideal was France, not democracy or the Republic. This made other political parties wary of allying with him, since they were not convinced he would keep France under a democratic government.

His convictions also meant that, in his mind, he should remain above political parties to gather all the French people. He did hesitate a lot before creating a party, and even when he did, he tried to make it into a ‘movement’ rather than a party, in the hope of gathering supports from other parties. It hindered the party he created, the RPF, which never managed to create a government during this 4th Republic despite some initial successes. The RPF was dissolved in 1955.

He still remained a popular figure, and he built an image of a kind of Cincinnatus, a potential recourse for the country. Then, in the 1950s the instability of the 4th Republic came to a peak due to the war in Algeria. All governments failed to solve this crisis, and things came to the point where the French Army was threatening the country with a putsch in 1958. To some extent, the French Army was actually wishing for De Gaulle to take power, but De Gaulle tried to keep some distance so as to appear as a legal, republican recourse. The President of the Republic of that time, René Coty, supported the idea and nominated De Gaulle as ‘Président du Conseil’ (Prime Minister) with exceptional powers and the mission to draft a new constitution.

This constitution has lasted to this day with a few modifications, and this was the beginning of the 5th Republic. After it was enacted, De Gaulle was elected President of the Republic in 1958 and again in 1965. After the instability of the 4th Republic, the great stability of the institutions of the 5th Republic is one of the reasons that De Gaulle has remained the most popular politician to date in France’s history. He also quashed another putsch in Algiers and negotiated independence for Algeria, solving the never-ending crisis and protecting democracy.

In may 1968, there was a major student strike against his government, which labor unions started following, threatening to degenerate into a new revolution in France. The reasons for this massive movement are still being debated; among other things, France under De Gaulle remained conservative (contraception had only been legalized in 1967), very catholic, and rather authoritarian. The rapid economic development of society, the increase in the number of students and the general social and cultural movements in France as well as in the rest of the world meant that many people, in particular students, rejected traditional authority. To answer your particular point, these students were born after 1945 and had no memories of De Gaulle’s leadership of the Free French, only as a conservative and authoritarian far from their own ideals.

This crisis shook the regime. De Gaulle went away for a state visit to Romania, then flew away to Germany, and showed himself unable to manage the crisis. However, the government under the leadership of the Prime Minister, Pompidou, managed to retain control of the situation through a mix of police repression, negotiations on salaries with the workers, and the call of legislative elections that ended in a triumph for the Gaullists.

A few months after this crisis, De Gaulle, who knew his time was short, decided to call a referendum on an important reform of the Republic, which included the creation of new regions and the merger of the Senate with another assembly. As he lost, he decided to resign because he considered that the exercise of power was no longer possible in these conditions.

It has often been interpreted as an excuse to leave power after the revolts from the prior year. The link, however, is not proven. But it is also very possible that De Gaulle, now aged 79, realized he no longer had the energy to lead the country, and that he was out of touch with the population. This is however more interpretation than fact. He certainly was very old and retired from politics. After a controversial trip to Franco's Spain, he passed away a year and a half after retiring.

I hope it helps to answer your questions. My main source is Julian Jackson’s biography of De Gaulle, unsurprisingly titled ‘De Gaulle’, that I can only recommend.