April 21 was the date of the Parilia (or Palilia), a festival of the goddess Pales that was originally about purifying fields and livestock for the upcoming agricultural season. Ovid, for example, describes the rites that were celebrated both in the city and in the countryside, which involved, among other things, jumping over fires. In his work on Roman festivals, the Fasti, he lists many possible origins for the festival, but concludes

“...is it not closer to the truth that, when Rome was founded, orders were given that the household gods be transferred to the new houses; and that, in changing their homes, the farmers set fire to their country houses and the cottages they were about to leave and leapt through the flames with their cattle too? It is a practice that continues even now on your birthday, Rome.” (Religions of Rome, vol. 2, pg. 118)

The association between the two dates probably goes back much further than Ovid's time int he 1st century AD, probably to Rome’s very early republican period, maybe even earlier. The association of the two dates is also mentioned by Plutarch in his Life of Romulus:

“Now it is generally agreed that the foundation of the city took place eleven days before the Kalends of May [21 April]. And this day is celebrated by the Romans with a festival, which they call the birthday of their country. In the beginning, so it is said, they sacrificed no living creature - but thought that they should keep pure and bloodless the festival commemorating the birthday of their country. However, even before the city's foundation, they had a herdsmen's festival on that day and they called it 'the Parilia’.” (Religions of Rome, vol. 2, pg. 119)

Plutarch goes on to cite some other ancient authors about how the actual year was calculated…it involves comparisons with the Greek and Egyptian calendars and it’s pretty confusing! But it should be enough to note here that the date comes out to 753 BC, according to our modern calendar. Plutarch attributes the calculation to the Antiquities of Marcus Terentius Varro, a Roman scholar from the 2nd century BC. Modern historians often call this the “Varronian system” or the “Varronian chronology”.

Varro’s Antiquities doesn’t survive today, but other ancient authors quoted from it, especially a 3rd-century writer named Censorinus. He noted that the year 239 (well he didn’t call it that, but the year 239 in our calendar) was 991 years since the foundation of Rome. By that time in the imperial period, the Parilia was no longer the ancient agricultural purification festival it had once been - it had evolved the Romaea, the celebration of Rome’s birthday and by extension the foundation of the Empire. In 48 AD, Emperor Claudius had celebrated the 880th anniversary of the founding of Rome, and in 148 Antoninus Pius had celebrated the 900th anniversary. Likewise, on April 21, 248, Emperor Philip celebrated Rome's millennium with lavish games and celebrations.

So, April 21 was the traditional date, coinciding with the Parilia, probably long before the Romans started writing about their own history. The year 753 BC was calculated by Varro in the 2nd century BC and was gradually adopted after that.

Sources:

Alan E. Samuel, Greek and Roman Chronology: Calendars and Years in Classical Antiquity (Munich, 1972)

Allen M. Ward, Fritz M. Hechelheim, Cedric A. Yeo, A History of the Roman People, 4th ed. (Prentice Hall, 2003)

Mary Beard, John North, and Simon Price, Religions of Rome, Volume 1: A History and Volume 2: A Sourcebook (Cambridge University Press, 1998)