The failed coups of 1962 (22nd February) and 1963 (20th May) were indeed led by Talat Aydemir, supported by his friends in the Army, and cadets in the army academy. The coups followed the successful coup d'etat of 27 May 1960 when the government of Adnan Menderes and the President Celal Bayar, both from the Democratic Party, were toppled.

The 27th May coup d'etat was conducted, as the military officers (and a large section of people, intellectuals, academics, etc.) as an attempt to re-establish liberal democracy in Turkey. They installed the new constitution of 1961, commonly seen as the most liberal one Turkey has ever had. Then, in the elections of 1961, the Justice Party, which was pretty much the continuation of the DP with lesser and more soldier-friendly faces, took 34% of the vote and came second. This, together with other signs of softening, strengthened the belief among some army figures that the coup was about to be reverted. Did they overthrow the Democrats only to re-install them to follow the same policies with an interval of a year?

Some, like Aydemir, thought no. First, there were 'the Fourteen'; fourteen officials (colonels and below) that had insisted on holding onto power a bit longer to conduct the reforms they found necessary in social and economic spheres, rather than directly leaving the power to the JP or the Republican People's Party. However, the generals managed to disperse with them in November 1960 by sending them on diplomatic missions abroad. These were all figures who played their roles in the coup of 27th May. By contrast, unlucky Aydemir had missed the coup because he was part of the Turkish army in Korea. Aydemir apparently had contacts with some left-Kemalist (as opposed to right-Kemalist or simply Kemalist) circles like the journal of Yön. He believed that establishing the same sort of democracy that led to the rise of Menderes would create the very same results: a country economically, socially, and internationally ruined. He, therefore, believed that the Army had to step in and conduct these reforms, much as the Yön circle.

He had two attempts. Aydemir was the Head of the Military Academy in Ankara (a post the generals would very much regret giving him later) and was thus able to establish a very good relationship with the cadets. On 22nd February 1962, he revolted in the Academy, though he was convinced to back off upon seeing that his forces were not enough to push for a serious change. Along with some of his friends, Aydemir was then retired, being saved from a more serious punishment by İsmet İnönü, the former President of Turkey, second-in-command to Mustafa Kemal Atatürk during his term and during the Turkish War of Independence. Aydemir tried his luck again on 20th May 1963. Again, he was unable to overthrow the government, with significant portions of the military, and most crucially the Air Force siding against him. He was arrested and, having spent all his chances of survival at the first attempt, he was condemned to death. He was executed later in 1964.

In Aydemir's failure, a key factor was İnönü. As İnönü was a national hero, his charisma among the cadets and on the army as a whole was too strong for Aydemir to succeed in open confrontation with 'İsmet Pasha'. It is true that the upper ranks were suspicious of the JP enough to sign a protocol in late October 1961 wherein they swore to do a second coup to prevent the JP from rising to power. In fact, it was due to negotiations between them and politicians including İnönü (then head of the RPP) and Ragıp Gümüşpala (the leader of the JP and a former general himself) that Turkey would have its first coalition government led by İnönü. Yet, the generals were also worried about extremely revolutionary groups among younger officials. If they were unhappy about the JP and if the worry about the younger officials would lead them to do something about the JP, they were not quite prepared to actually do a second coup. With İnönü, other political parties, a good portion of the armed forces and certainly the higher command being against him, honestly, Aydemir never had much of a chance. He could have won only if he was prepared to spill more blood than he actually did. He was not and hence he lost.

So, Turkey went on to have a combination of coalition governments from 1961 to 1965 until the rise of a gentleman called Süleyman Demirel, who first became the chairman of the JP and then won the elections of 1965 to become the prime minister. Turkey had some of its most politically vibrant (in both good and bad senses of the word...) between 1961 and 1971 largely due to relatively liberal policies of Demirel at the time and the new rules established by the 27th May regime. This lasted until 12th March 1971 when a new coup d'etat, once more, changed much (though not enough: it took a further coup in September 1980 to create a new order in Turkey). The coup of 12th March is a completely different story, belonging to another day...

TL;DR: Talat Aydemir believed that the coup of 27th May 1960 was being betrayed by politicians and generals alike. He thought a new coup was needed to reform Turkey socially and economically before re-establishing democracy. He tried twice on 22nd February 1962 and 20th May 1963. He failed twice, finding himself in retirement following the first but being arrested and ultimately sentenced to death after the second. His sentence was carried forward in 1964 and a year later, Turkey entered a new era under the rule of Süleyman Demirel, the new leader of the JP.

The source (and the recommendation!) for this comment is Feroz Ahmad, The Turkish Experiment in Democracy: 1950-1975 (London: C. Hurst & Company, 1977), especially pp. 176-191.