What does the worshiper say after each takbir in the Eid prayer?

No specific remembrance (dhikr) between the Eid takbirat (exclamations of 'Allah Akbar') has been preserved from the Prophet (peace be upon him). However, some Salaf (early Muslims) have narrated practices, and scholars differ on the matter: Abu Hanifa, Malik, and Al-Awza'i held the view of continuous takbir without any dhikr in between, while Ash-Shafi'i and Ahmad maintained the permissibility of dhikr between every two takbirat. They based their argument on the actions of Abdullah ibn Mas'ud (may Allah be pleased with him), as narrated by Al-Athram and Al-Bayhaqi, where he would praise Allah, send blessings upon the Prophet, and supplicate between the takbirat.

The matter is broad. There is no harm for one who remains silent or for one who remembers Allah. Remembrance is preferable due to its establishment from Ibn Mas'ud, and no other Companion opposed him on this. The reported dhikr includes praising Allah, extolling Him, sending blessings upon the Prophet (peace be upon him), and supplicating. Alternatively, one may say, "Subhan Allah, wal-hamdulillah, wa la ilaha illa Allah, wa Allahu Akbar" (Glory be to Allah, and praise be to Allah, and there is no god but Allah, and Allah is the Greatest), and send blessings upon the Prophet and supplicate. Or, one may say, "Allahu Akbaru kabira, wal-hamdulillahi kathira, wa Subhan Allahi bukratan wa asila" (Allah is the greatest, greatly; much praise be to Allah; and glory be to Allah morning and evening). There is no specific dhikr fixed from the Prophet (peace be upon him) or the Companions.