Why Do We Say Bismillah before Any Task?
Question

My question is about the meaning of the phrase Bismillah and the reason for starting actions with it. Is it permissible, according to Islamic teachings, to explain it as follows, noting that this is based on my own understanding: The phrase Bismillah is said as if we are doing things on behalf of Allah, and that is because we humans are vicegerents of Allah on earth, and we do things in the dominion of Allah as we are appointed as vicegerents by Him – {and for Allah is the highest attribute} [An-Nahl 16:60] – and likenesses are given to explain, not by way of analogy. If a man speaks on behalf of a friend of his called `Ali and says: I am speaking to you on behalf of my friend `Ali, who is demanding such and such, and here we find that the man is speaking on behalf of someone on whose behalf he speaks and makes demands using his name, then is it valid to say that when we eat food, for example, and say Bismillah, that we are representatives of Allah? What other meaning [of Bismillah] is there, if there is one?
Praise be to Allah.Meaning and Significance of Bismillah
The meaning of Bismillah is: I begin my actions, seeking the help of Allah and seeking blessing (Barakah) by beginning in His name.
In the phrase Bismillah (lit. In the name of Allah), there is something which is omitted, namely: I begin my action in the name of Allah. So saying Bismillah before reading the Quran means: In the name of Allah I read. Saying Bismillah before writing means: In the name of Allah I write . And likewise, one means: In the name of Allah I ride, in the name of Allah I eat , and so on.
Ibn Qutaybah said in Gharib Al-Quran (1/51): Sibawayh said: What is meant is: I begin in the name of Allah or I start in the name of Allah. (End quote)
Contextual Application of Bismillah
Shaykh Muhammad ibn `Uthaymin (may Allah have mercy on him) said:
The phrase Bismillah is a prepositional phrase (Jarr wa Majrur) that is connected to an implied action that is appropriate to the context. So when you want to read the Quran, it implies: In the name of Allah I read. When you want to do Wudu, it implies: In the name of Allah I do Wudu. When you want to slaughter an animal, it implies: In the name of Allah I slaughter. 
Why do we say Bismillah before any task?
We regard what is implied as a verb, because there is an implicit action, and we say that the action is implied after saying Bismillah for two reasons:

Seeking blessing (Barakah) by beginning in the name of Allah, may He be Glorified and Exalted.
To make our beginning exclusively in the name of Allah.

We say that this is an implicit, specific action because this makes the meaning clearer.  For example, if you said, when wanting to read a book: In the name of Allah I begin, it is not known what you are beginning to do. But “In the name of Allah I read” is a clearer indication of what you intend to begin." (End quote from Ash-Sharh Al-Mumti`,  1/7)
Avoiding Misinterpretations of Bismillah
It is not appropriate to say that what it means is doing something on behalf of Allah. The context in which we say Bismillah indicates that it is not possible to say such a thing about Allah. Hence it is not appropriate to say that someone is eating on behalf of Allah, or doing such and such on behalf of Allah. Exalted be far above that.
Even with regard to matters of religion, the teachings of Islam and what Allah has told us, Allah did not give anyone the right to speak on His behalf or explain His religion and laws on His behalf except the Prophets (peace be upon them). They are the ones who convey from Allah His revelation and His laws to His slaves.
Everyone apart from the Prophets only speaks on behalf of his own self, and decides on the basis of his own understanding and the level of knowledge that he has attained. He does not speak in the name of Allah, may He be Glorified and Exalted.
In Sahih Muslim (1731), it is narrated from Sulayman ibn Buraydah, from his father who said:
When the Messenger of Allah (blessings and peace of Allah be upon him) appointed commanders for an army or expedition, he would advise them personally to fear Allah and to be good to those of the Muslims who were under their command. Then he said: “… If you lay siege to a stronghold, and the people ask you to promise them the protection of Allah and His Prophet (blessings and peace of Allah be upon him), do not give them the promise of the protection of Allah and His Prophet; rather give them your promise of your protection and that of your companions, for then if you break your promise and that of your companions, that is less serious than breaking the promise of Allah and His Messenger. If you besiege a stronghold and the people want to make a deal on the basis of the ruling of Allah, do not make a deal on the basis of the ruling of Allah; rather make a deal on the basis of your own ruling, for you cannot be certain whether you will be able to work out a deal with them that is in accordance with Allah’s ruling or not.”
For more details, please see the following answers: 22186 , 173672 , 175312 .
And Allah knows best.