Meaning of Labbayk Allahumma Labbayk
Question

“Labbayka Allahumma labbayk, labbayka la sharika laka labbayk. Inna al-hamd wa’l-ni’mata laka wa’l-mulk, la sharika lak (Here I am, O Allah, here I am. Here I am, You have no partner, here I am. Verily all praise and blessings are Yours, and all sovereignty, You have no partner).”
 This is the Talbiyah recited by the pilgrim doing Hajj and ‘Umrah. What is its meaning and what is the benefit of it?
Praise be to Allah.Hajj: A symbol of Tawhid
Hajj is the symbol of Tawhid (the Oneness of Allah) from the first moment the pilgrim enters ihram. Jabir ibn ‘Abd-Allah said, describing the Hajj of the Prophet (peace and blessings of Allah be upon him): “Then he started to say the words of Tawhid, ‘“Labbayka Allahumma labbayk, labbayka la sharika laka labbayk. Inna al-hamd wa’l-ni’mata laka wa’l-mulk, la sharika lak  (Here I am, O Allah, here I am. Here I am, You have no partner, here I am. Verily all praise and blessings are Yours, and all sovereignty, You have no partner).’” (Narrated by Muslim)
Anas said, describing how the Messenger of Allah (peace and blessings of Allah be upon him) entered ihram, that he said: “Labbayka ‘Umratan la riya-a fiha wa la sum’ah (Here I am for ‘Umrah in which there is no showing off or seeking reputation).”  
This trains the soul to acknowledge the Oneness of Allah and to be sincere towards Him. 
The pilgrim starts his Hajj with Tawhid, and continues to recite the Talbiyah with its words of Tawhid, and he moves from one action to the next with Tawhid . 
Meaning of the Talbiyah
The Talbiyah has a number of meanings, such as: 

“Labbayka Allahumma labbayk (Here I am, O Allah, here I am)” is one response after another, which is repeated to show that the response is lasting and ongoing.
“Labbayka Allahumma labbayk (Here I am, O Allah, here I am)” means, I submit and submit again, i.e., ongoing submission.
It may mean staying in one place (labba bi’l-makan) and clinging to it, which means, I am persisting in obeying You, thus emphasizing the meaning of continuous servitude to Allah.
One of the meanings of the Talbiyah is confirming the love of Allah. There is an Arabic phrase imra-ah labbah (a loving woman) which refers to a woman who loves her child. One only says “Labbayk (Here I am at your service)” to a person whom one loves and respects.
It implies sincerity, as in the phrase lubb al-shay which means the essence of a thing, and lubb al-rajul which means a man’s mind and heart.
It implies drawing close, as in the word ilbab, which means drawing close, so it emphasizes the meaning of seeking to draw closer and closer to Allah.
It is a symbol of the Tawhid of the religion of Ibrahim , which is the spirit and aim of Hajj, indeed the spirit and aim of all the acts of worship. Hence the Talbiyah is the key to this act of worship that the pilgrim is embarking on.

The Talbiyah also includes: 

Praise of Allah, which is the dearest thing with which a person may draw close to Allah. 
It implies recognition of the blessings of Allah, for He is the source of all blessings and the One Who bestows them. 
And it includes acknowledging that all sovereignty belongs to Allah Alone, and no one else has any true dominion. (See Mukhtasar Tahdhib al-Sunan, by Ibn al-Qayyim, 2/335-339) 
When the pilgrim is reciting the Talbiyah, he feels a connection with all other created beings, as they all join with him in submission to Allah Alone and echo his Talbiyah. The Messenger (peace and blessings of Allah be upon him) said: “There is no Muslim who recites the Talbiyah but whatever is to his right and to his left of stones, rocks and clods recites it with him, to the furthest point to the east and the west” – meaning from his right and his left. (Narrated by al-Tirmidhi, 828; also by Ibn Khuzaymah and al-Bayhaqi  with a sahih isnad.)

And Allah knows best.