Salam Dear Elizabeth,
Thank you for your question and for contacting Ask About Islam.
Let us quickly review the background of the chant you mentioned; it is called Takbir of Eid. Eid means an Islamic public celebration. Such celebrations occur twice a year, each time following the completion of one of the five pillars of Islam.
The first, Eid al-Fitr, marks the completion of fasting the blessed month of Ramadan, which is the fourth pillar of Islam. The second is Eid al-Adha, and marks the completion of Hajj (Pilgrimage to Makkah), which is the fifth pillar of Islam.
According to the Sunnah (tradition) of the Prophet (peace and blessings be upon him) everyone is urged to attend Eid Prayer—men, women, and children. During Eid, Muslims express their gratitude to Allah for enabling them to observe the major acts of worship preceding each Eid. They also visibly show joy and congratulate each other. Therefore, Eid is a day of joy, thanksgiving to Allah, brotherhood, and unity.
Muslims have no public celebrations apart from Eid al-Fitr and Eid al-Adha. The Prophet (peace be upon him) said what means:
"...Every nation has its festival, and this is your festival." (Al-Bukhari)
Here, he referred to the fact that these two Eids are exclusively for the Muslims.
The time for Eid Prayer is when the sun has risen about three meters (10 feet) above the horizon. Eid Prayers cannot be said after midday. It is better to hold Eid Prayers in the open if there is no rain or bad weather. The Prophet never performed Eid Prayers in the mosque except once due to rain.
There is neither Adhan (call to prayer) nor Iqamah (second call to prayer) for these prayers, which consist of two rak`ahs with twelve takbirs (Allahu Akbar); seven in the first rak`ah and five in the second. After performing prayer, the imam delivers a khutbah (sermon, speech) just like Friday khutbah.
Here Are Some of the Good Manners of Eid:
It was reported that Sa`id ibn Jubayr said, "Three things are Sunnah (prophetic traditions, preferred to be followed by Muslims) on Eid: to walk (to the place of prayer), to take a bath, and to eat before coming out (if it's Eid al-Fitr)." People should exchange greetings on Eid. They walk to prayer wearing their best clothes and change their route on returning.
It is also considered a good Islamic practice to visit one another and exchange gifts. The Prophet (peace be upon him) said:
"Exchange gifts in order to foster love." (Al-Bukhari in his book Al-Adab Al-Mufrad)
The true spirit of Eid is reflected in generosity to the poor and the needy. It is recommended to make lots of istighfar (asking Allah for forgiveness) and pray that Allah accept our fast, pilgrimage, prayers, and devotion, and grant the Muslim nation glory and success.
The loud chanting of Takbir of Eid is one of the greatest sunnahs of this day. It brings us to your question of the meaning of the Takbir. The transmitted wording of takbir is: "Allahu Akbar, Allahu Akbar, la Ilaha illa Allah; Allahu Akbar, Allahu Akbar, walillah al-hamd".
There are other forms used by the Prophet's Companions and reported to us from trusted chains of transmission. They are generally combinations of the expressions below, which are the most respected forms of praise and exaltation to Allah in Prophetic traditions.
They are to be recited loudly and in unison by everyone attending Eid Prayer and everyone on their way there. This show of power enforces the feeling of unity and solidarity and declares the joy of Eid to all. Here are the meanings of the expressions used.
Allahu Akbar: This statement is said by Muslims numerous times. During the call for Prayer, during Prayer, when they are happy and wish to express their approval of what they hear, when they slaughter an animal, and when they want to praise a speaker.
It means "Allah is the Greatest." Muslims praise Allah in every aspect of life, and as such, they say "Allahu Akbar" as they proceed to Eid Prayer.
La ilaha illa llah: This expression is the most important one in Islam. It is the declaration of faith that every person has to announce to be considered a Muslim. It is part of the first pillar of Islam.
It means "There is no god worthy of worship except Allah." The second part of this declaration is to say "Muhammadun Rasul Allah," which means "Muhammad is the Messenger of Allah."
Al-hamdu lillah: This is an expression from the Quran that Muslims recite and say many times per day. Other than being recited daily during prayers, a Muslim recites this expression in every activity of his daily life.
It means "all praise be to Allah" and is said as an expression of thanks and gratitude to Allah. A Muslim (a person who chooses to submit peacefully to the Creator) invokes the praises of Allah when he finishes any work, thanking Allah for His favors.
A Muslim is grateful to Allah for all His blessings. It is a statement of thanks, appreciation, and gratitude from the Creature to his Creator.
Subhan Allah: This is an expression that Muslims use to express wonder, exclamation, or surprise; it is also an expression of respect for Allah.
It means: "Allah is Exalted. Glory be to Allah. Far removed is He from anything imperfect associated with Him and from anything unsuitably ascribed to Him."
Allahumma salli `ala Muhammad: This is an expression that Muslims use whenever the name of Prophet Muhammad is mentioned or written. It means "may the blessings and the peace of Allah be upon Muhammad."
In the Quran, Allah has ordered Muslims to say such an expression. This reflects the Muslim's love and affection towards the Last Prophet, who carried the burden of conveying the last divine message to mankind.
Allahu Akbaru Kabeera: This expression is simply an emphasis on the greatness of the Creator. It means that God is Greater than everything that exists and the term "kabeera" is a superlative form of the same meaning.
Al hamdu lillahi katheera: This is also an expression of emphasis on the gratitude of the Muslims towards their Creator. It's meaning is: "high gratefulness and thankfulness is to be to Allah, the Creator."
It is strongly recommended to repeat in a loud voice Allahu Akbar, Allahu Akbar, La ilaha illa Allah, Allahu Akbar wa lillahi-l-hamd after every congregational prayer on the days and nights of Eid.
I hope this answers your question. Thank you and please stay in touch.