O Allah! You are Salam

بسم الله الرحمن الرحيم

The Prophet of Allāh sallallāhu 'alayhi wa sallam would say at the end of every prayer, اللَّهُمَّ أَنْتَ السَّلَامُ، وَمِنْكَ السَّلَامُ، تَبَارَكْتَ يَا ذَا الْجَلَالِ وَالْإِكْرَامِ (O Allāh! You are As-Salām and from you is as-salām. Most Blessed are You, O Owner of Majesty and Honour)” [Ibn Mājah].

O Allāh! You are As-Salām
As-Salām is one of Allāh’s Names. The word salām linguistically comes from the root sīn-lām-mīm which means absence or safety from any number of unfavourable things. Accordingly, As-Salām as a Name of Allāh denotes One who is Serene in His Existence, Beautiful in His Being, Impeccable in His Conduct and Flawless in every way because He is high above any susceptibility to harm, free of every deficiency, and innocent of any fault or indecorum. 

This completion or absence of imperfection (salāmah) is witnessed in all of Allāh’s Attributes. For example, He is Al-'Alīm (Knowing) without the imperfection of ignorance and forgetfulness. He is Al-Hayy (Living) without the imperfection of death and ill health.

And from You is as-salām
This is a type of implicit supplication (du‘ā) for completion, serenity and beauty, which is elaborated further below in the article.

A supplication is worded either explicitly or implicitly. 
 
An example of an implicit supplication is the du‘ā of Ayyūb 'alayhis-salām, "And (remember) Ayyūb when he called out to his Lord, 'Indeed, harm has touched me and You are the Most Merciful of those who show mercy'" [21:83].
 
There is no explicit request for relief, even though that is what he intended. He could have explicitly said, "I have been bereft of my wealth, my health and my family. Relieve me! O Allāh! Return to me to what I had lost."
 
Similarly, Yūnus alayhis-salām intended rescue in the belly of the whale, saying, "There is no deity but You, Glorified be You! Truly, I have been of the wrongdoers" [21:87] and Allāh answered, "So, We responded to him and saved him from the distress. And thus do We save the believers" [21:88].

The prophets in both examples intended relief with implicit phrased duas.

Most Blessed are You, O Owner of Majesty and Honour
This is a reiteration of Allāh’s perfect condition. He is blessed, majestic and honourable in every way.
 
The implicit request
Upon finishing the prayer, the Muslim makes this request (dua) after the istighfār (astaghfirullāh x3). Through the istighfār he seeks forgiveness for the heedlessness and shortcomings in the salāh, whilst through “Allāhumma antas-Salām…” he intends completion, beauty and perfection in the next (i.e. absence of flaws and blemishes). This is his mindset for the succeeding prayer [and all his self and deeds].

In other words, "O Allāh! You are Serene, Complete and Beautiful in every way (As-Salām). Make my prayer serene, complete and beautiful. Only then can it be worthy of being directed towards You because You are Perfectly Beautiful and You alone deserve beautiful worship."

والحمد لله رب العالمين
وصلى الله وسلم على نبينا محمد