A lot of it was the result of foreign influence from Western Europe. The British intervention in Greece had a desire to restore the home of the philosophers who founded Western Civilization. They extrapolated the history of Ottoman-occupied Greece to the Classical Age. This was in many ways a result of the virulent neo-classicism present in late 18th to early 19th century Europe, which sought to emulate old, Republican Rome and not Imperial Rome. In this mindset, Greece was considered the land of thinkers and philosophers, and not the place where the Roman Empire survived for an additional thousand years.

Initially, Greeks actually sought to restore the Roman Empire, as exemplified by early independence organizations like the Filiki Eteria. Additionally, Constantine XI was used as a sort of martyr figure by revolutionaries, who sought to restore continuity between their efforts and Rome Indeed, some vestiges of later Rome are entirely visible in the Greek Kingdom, one need only look to the unofficial numbering of their Constantines (XII and XIII). Furthermore, the Megali Idea, although adopting many Panhellenic ideals, was in many ways a Roman idea, with its main objective being the retaking of Constantinople as its capital, not Athens. Upon the birth of the first, Greek-born prince (later king), the people supposedly demanded he be called Constantine XII, as he would be destined to retake Constantinople. Eventually, they settled on officially calling him Constantine I. Still, many actually believed him to be the reappearance of Constantine XI Palaeologus, who never died and was merely waiting to be reawakened.

EDIT: I must make a distinction between the popular narrative and the political history of Constantine I's reign. The Megali Idea was actually propagated by radical Greek nationalists and republicans who sent the king into exile.

The British (as well as the French and other allies) had a lot to do with it during the Greek revolution. As the Greeks revolted, the western nations aided them in their revolution and after the revolution was successful, they helped Greece rebuild.

Before they completely fell to the Turks, the capital was Constantinople, and then eventually Mystras as the empire collapsed. During the revolution the capital was located in Corinth, and after the revolution, they relocated the capital to Athens.

Part of this relocation to Athens (which was a farming village by that point) was part of a redefinition of Greek identity to how western europe saw Greece, rather than how the Greeks saw themselves.

So they ended up completely designing Athens after western neoclassical style of urban planning. The Acropolis itself had many buildings on it, dating obviously from the classical era, to the Roman Empire, to the 19th Century. The western nations in helping to rebuild and redefine Greece, had all buildings on the Acropolis completely levelled except for the classical ones, so what you see today is, at least according to their narrative, what it would have looked like several thousand years ago in classical Greece.

In fact, even in the reconstruction of Orthodox Churches, a lot of them were rebuilt according to western architectural styles at the time rather than according to the Byzantine styles.

This whole redefinition continued, even as the Greeks wanted to restore the Roman Empire. When the Russians fought the west and the Turkish Empire during the Crimean War, the Russians felt it was their duty to protect all Orthodox people and to aid all Orthodox people in revolutions against the Turks. The West wanted to contain Russia, and in the end, defeated Russia. Greece actually wanted to help Russia during the conflict and wanted to gain more territory from the Turks, but its western allies prevented it from participating.

Prior to World War I, Greece went to war against Turkey yet again, and advanced within just a few miles of Constantinople, had gained Smyrna and a large portion of the western coast of Turkey. Unfortunately a few military blunders led to them being pushed back to the coast when they neared Ancyra (Ankara). Eventually the western nations forced Greece to sign a treaty which required it to give back a lot of the territory it had gained, and they also forced Greece and Turkey to exchange populations, with Turkey sending the Greek Orthodox (except a few in Constantinople) to Greece, and Greece sending the Muslims to Turkey.

This was considered a national distaster for Greece. It almost completely destroyed any hope the Greeks had for regaining the territory lost when the Roman Empire fell.

Ever since then, and with the rise of the Olympics, the redefinition of Greek identity continued and many Greeks today see themselves more as "westerners" than people of the east.

To a degree, they still identify with their ancestors in the Roman Empire, and still identify with their Orthodox neighbors. During the recent conflicts over Yugoslavia, many Greeks actually travelled to Serbia to help Serbia fight against the Muslim populations.

What is called the "Great Idea" of recapturing territory lost 500 years ago is still somewhat alive. Many young Greeks don't really care and are mostly focused on their current political movements, finding gainful employment etc... In fact, many end up travelling to western countries like the UK to get employment or to be educated.

A lot of the older population of Greece still (somewhat) hold on to that idea though. And for the religious people of Greece, there are some prominent and popular prophecies by highly respected, venerated monastic elders in Greece that started 300 years ago and the most recent date to the late 1980s. Those prophecies basically predict that the territory will eventually be given back to Greece, but Greece won't be the ones to capture it, it will happen in a world conflict and Greece will have to stay neutral because of economic problems and a famine. There are many in Greece who hold to those prophecies still, but also many (especially the youth) who dismiss them.

Anyway, it basically started with the Greek Revolution, and has lasted to today, mainly due to Greece's close relations with allies in the west.