Ramadan is the month of Huda (guidance) and so as the takbirs of Eid: 

Dear noble brothers, saying ‘Allahu Akbar’ (Allah is the Greatest) at the first day of Eid, while obeying a creature and disobeying Allah ‘The Creator’ is useless; even if he repeated it thousands of times. 
In fact, you think that you obey the strong, but if you realized that Allah, Glory be to Him, has control over everything, you would never disobey Him.
So,

“And that you must magnify Allah [i.e. to say Takbir (Allahu-Akbar; Allah is the Most Great) on seeing the crescent of the months of Ramadan and Shawwal] for having guided you”

[Al-Baqarah 185]

Eids of Muslims always come after great acts of worship. Eid Al-Fitr comes after the worship of Siyaam (fasting), and Eid Al-Adha comes after the worship of Hajj (pilgrimage), the worship which has been imposed once in a lifetime.

Eid means that you have returned to Allah, and  reconciled with Him.

“And that you must magnify Allah [i.e. to say Takbir (Allahu-Akbar; Allah is the Most Great) on seeing the crescent of the months of Ramadan and Shawwal] for having guided you so that you may be grateful to Him.”

 [Al-Baqarah 185]

The purpose of the worship in Ramadan is to approach to Allah, The Almighty; so, 

“And when My slaves ask you (O Muhammad, PBUH) concerning Me, then (answer them), I am indeed near (to them by My Knowledge).”

[Al-Baqarah 186]

It means that Muslim performs Siyaam, Salah, Hajj and Zakat in order to approach to Allah, glory be to Him.