The Most Great, The All-Great, The Greatest.
The name Al-Kabeer (in Arabic: ٱلْكَبِيرُ) is the 37th name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ meaning the perfection of greatness. The heart of the believer knows Allah is the essence of perfection and nothing is comparable to him. His greatness is beyond measure, and His being is unimaginable to the human mind.
From the root kaf-ba-ra (ك ب ر), which has the following classical Arabic connotations: to be great in rank, dignity, nobility, majesty to be great in size, vast, formidable to be great in age, oldest to be great in learning, most knowing to have rights above all others.
In Arabic, the word kabir (كبير) is used by modern Arabic speakers to describe something great, big or large, often in a physical sense. The usage of the definite article "al" before the name is demonstrative of being THE biggest or THE greatest. But with Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ this is not limited in a physical sense. The name Al-Kabir is often used in tandem with Al-Aliyy (The Most High), likely to illustrate that Allah is great in all respects, greater than any mental constructs we may have of Him. It is a name of Allah's perfection, not meant in a comparative sense. He is greatness and is alone in His greatness.
Zaalika bi annal laaha Huwal Haqqu wa anna maa yad’oona min doonihil baatilu wa annal laaha Huwal Aliyyul Kabeer (section 3)
"That is because Allah is the Truth, and that what they call upon other than Him is falsehood, and because Allah is the Most High, the Grand." — (Qur'an 31:30)
Wa laa tanfa'ush shafaa’atu 'indahooo illaa liman azina lah; hattaaa izaa fuzzi'a an quloobihim qaaloo maazaa qaala Rabbukum; qaalul haqq, wa Huwal Aliyul Kabeer
"And intercession does not benefit with Him except for one whom He permits. [And those wait] until, when terror is removed from their hearts, they will say [to one another], “What has your Lord said?” They will say, “The truth.” And He is the Most High, the Grand." — (Qur'an 34:23)
Aalimul Ghaibi wash shahaadatil Kaabeerul Muta’aal
"[He is] Knower of the unseen and the witnessed, the Grand, the Exalted." — (Qur'an 13:9)
Expanding on the meaning of Allahu Akbar:
The name Al-Kabir appears a total of six times in the Qur'an. It's also an attribute of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ you're used to hearing on an everyday basis. In the call to prayer (i.e., the adhan), we say Allahu Akabar meaning "Allah is greater." This contains the importance of salah time, to stop all that you're doing when the adhan comes on, and hayya alas salah, meaning "come to prayer." Why? Because God is greater - He is greater than whatever else may be preoccupying your time. Here it seems to be used in the comparative sense, but really it's an affirmation of Allah's absoluteness.
Although the reference to the last verse of Surah Jumuah is about Jumuah prayer, the message remains true for all your prayers, "O you who have believed, when [the adhan] is called for the prayer on the day of Jumu’ah [Friday], then proceed to the remembrance of Allah and leave trade. That is better for you, if you only knew." It's only befitting that the next part in the adhan mentions hayya alal falah "come to success." Narrated `Abdullah: I asked the Prophet (ﷺ) "Which deed is the dearest to Allah?" He replied, "To offer the prayers at their early stated fixed times." 
Al-Kabir - testament of tawhid:
Upon realizing only Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ possesses true greatness, we, therefore, praise Him alone. Only Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ has true worth. Everything else is perishable and exists in a temporary state. The one who is only conscious and glorifies himself lives in denial of this truth. Consciousness of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ and His greatness affirms His divine oneness and allows a person to live beyond his selfish desires and pursuits. It creates the space for detachment from oneself and to live for Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ, and Allah knows what's best for His creation.
Zaalika bi annal laaha Huwal haqqu wa anna maa yad’oona min doonihee huwal baatilu wa annal laaha Huwal Aliyyul kabeer
"That is because Allah is the Truth, and that which they call upon other than Him is falsehood, and because Allah is the Most High, the Grand." — (Qur'an 22:62)
This name is also a reminder to check our egos and to be humble because we know Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is far superior. Those who directly challenge Him will be among the losers. It was narrated from Abu Hurairah that the Messenger of Allah ﷺ said: “Allah, the Glorified, says: ‘Pride / Grandeur is My cloak and greatness My robe, and whoever competes with Me with regard to either of them, I shall throw him into Hell.’”  Even those who don't directly challenge Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ but are narcissistic and have a delusional sense of self-worth - they may bully, humiliate and berate others but will be faced Allah's vengeance. We are all created equal in the sight of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. The better one is determined by moral excellence. In the Qur'an, Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ says, "O mankind, indeed We have created you from male and female and made you peoples and tribes that you may know one another. Indeed, the most noble of you in the sight of Allah is the most righteous of you. Indeed, Allah is Knowing and Acquainted." (Qur'an 49:13)
Mu'sab b. Sa'd reported on the authority of his father that a desert Arab came to Allah's Messenger ﷺ and said to him: Teach me the words which I should (often) utter. He said: Utter, 
la 'ilah 'iila alllah wahdah la sharik lah alllah 'akbar kabiran walhamd lillah kathiran subhan alllah rabb alealamin la hawl wala quwwat 'iila bialllah aleaziz alhakim
"There is no god but Allah, the One, having no partner with Him. Allah is the Greatest of the great and all praise is due to Him. Hallowed be Allah, the Lord of the worlds, there is no Might and Power but that of Allah, the All-Powerful and the Wise."
The Greatness within you:
Another way a believer benefits from knowing Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Kabeer is acknowledging and respecting the beauty that is you. If we understand the superlative that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is the highest and greatest then it must be that everything that comes from Him must be perfect and great in its own respect. Whether or not we recognize and accept our beauty and our greatness is a different question altogether. To begin transforming self-loathing to self-loving has to do with reprogramming the story that's been conditioned in your psyche and reworking it. It takes some believing and trust to attach a new story to take its place, and whose word is greater to follow and trust than Al-Kabeer?
In the Qur'an, Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ mentions His creation as being molded perfectly, "It is Allah who made for you the earth a place of settlement and the sky a ceiling and formed you and perfected your forms and provided you with good things. That is Allah, your Lord; then blessed is Allah, Lord of the worlds." (Qur'an 40:64) and also, "We have certainly created man in the best of stature;" Qur'an 95:4 Would you refute Him, saying, "look, I respect your opinion but I am objectively ugly. See my left eye is slightly bigger than my right!" It may seem ridiculous, but we are all guilty of doing this in some fashion. This is probably even more exaggerated today because of the high beauty standards pushed on by social media.
Another beautiful ayat from the Qur'an worth deeply reflecting on is, "You now see the mountains and consider them firmly fixed, but then they shall pass away even as clouds pass away. That will be the handiwork of Allah Who has created everything with perfect wisdom. He is well aware of what you do." (Qur'an 27:88) The commentators of the Qur'an interpret "pass away" as a reference to the Earth itself, meaning that the world, believed to be solid, is actually as ephemeral as a cloud. It is temporary.
We can even try to extrapolate from this ayah on a personal level. Things that may seem firmly fixed will pass. If we can accept this, we can learn to let go of any emotional attachments. There is nothing that exists with absolute permanence but Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. Holding on to what was or is, is an impossible task. The one who caused the change is Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ, and He is the one who has perfect wisdom.
We can experience inner peace only with acceptance of what He has willed. If we're flexible with changing events, we can see the many blessings Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ has given each one of us. Try to drop the anxiety and self-criticism if you're not doing what you feel you're supposed to be doing - or where you think you should be. Harsh criticism of yourself serves no purpose but to hinder future progress. Instead, just focus on the smallest step you can take to improve your situation. Just as Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ spreads His perfection through His creation, we work to perfect our inner character. Emotional temperance, knowledge, deen, love, family, friendships, etc. Whatever else happens is up to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ, and we know that He is Al-Kabeer, The Greatest.
 Sahih al-Bukhari 527
 Hasan (Darussalam) Sunan Ibn Majah 4174 & Sahih (Al-Albani) Sunan Abi Dawud 4090
 Sahih Muslim 2696