The addressed is the Messenger Muhammad (saws) and everyone who follows him.

سَبِّحِ (Exalt): Glorify the Name of your Lord, the Most High by negating the existence of any counterpart to Him in actions, sayings, and attributes:

 {لَيْسَ كَمِثْلِهِ شَيْءٌ ۖ وَهُوَ السَّمِيعُ الْبَصِيرُ}

(There is nothing like unto Him, and He is the Hearing, the Seeing.) (Ash-Shura: 11)

When we read the verses which talk about glorifying Allah (swt) we find that Allah (swt) has been exalted since eternity, even before those who are glorifying Him were created. Hence, glorifying and exalting for Allah (swt) existed before the creation.

However, our praise to Allah (swt) does not benefit him, rather we are the beneficiaries.

The word “exalt” (glorify) in the Holy Qur’an comes sometimes in a past tense:

 { سَبَّحَ لِلَّهِ مَا فِي السَّمَاوَاتِ وَالْأَرْضِ ۖ وَهُوَ الْعَزِيزُ الْحَكِيمُ}

(Whatever is in the heavens and earth exalts Allah, and He is the Exalted in Might, the Wise..) (Al-Hadid: 1)

And sometimes it is mentioned in the present form:

{ يُسَبِّحُ لِلَّهِ مَا فِي السَّمَاوَاتِ وَمَا فِي الْأَرْضِ الْمَلِكِ الْقُدُّوسِ الْعَزِيزِ الْحَكِيمِ}

(Whatever is in the heavens and earth is exalting Allah, and He is the Exalted in Might, the Wise.) (Al-Jumu’a: 1)

So, praising to Allah (swt) do spreads in every place in this universe and every time; in the past, at present and in the future by everything and everyone:

 {وَإِن مِّن شَيْءٍ إِلَّا يُسَبِّحُ بِحَمْدِهِ وَلَٰكِن لَّا تَفْقَهُونَ تَسْبِيحَهُمْ}

(And there is not a thing except that it exalts [Allah] by His praise, but you do not understand their [way of] exalting.) (Al-Israa: 1)

Being (do not understand their way of exalting) indicates that it is a real glorifying since every creature has a certain way of glorifying that is understood by just those whom Allah (swt) has favoured and bestowed them of His light. Allah (swt) says about Dawud (Peace be upon him):

 {إِنَّا سَخَّرْنَا الْجِبَالَ مَعَهُ يُسَبِّحْنَ بِالْعَشِيِّ وَالْإِشْرَاقِ}

(Indeed, We subjected the mountains [to praise] with him, exalting [Allah] in the [late] afternoon and [after] sunrise.) (Saad: 18)

{ يَا جِبَالُ أَوِّبِي مَعَهُ وَالطَّيْرَ ۖ وَأَلَنَّا لَهُ الْحَدِيدَ}

(“O mountains, repeat [Our] praises with him, and the birds [as well].” And We made pliable for him iron,) (Saba: 10)

Meaning that the mountains are exalting Allah (swt) with Dawud (peace be upon him) and with others but what distinguished Dawud is that Allah (swt) has taught him the language of everything, thus, his praising agreed with the praising of everything and the birds.

Moreover, Allah (swt) has taught also Sulayman (peace be upon him) the language of ants and birds, so that he could speak to them and understand their speech.

Therefore, this verse:

 {سَبِّحِ اسْمَ رَبِّكَ الْأَعْلَى}

(Exalt the name of your Lord, the Most High, (1))

means that all the universe is glorifying Allah (swt), so, be in harmony with the universe around you and do not deviate from this system of praising.

The following verses are clarifying this exalting, as if saying “glorify the name of your Lord because he has done so and so”.

الْأَعْلَى (The Most High): Higher than everything and greater than everything. This word is a superlative adjective. Allah (swt) describes Himself as (The Most High) and describes some of his creation by (haughty), as such the cursed Iblis. Allah (swt) said to Iblis:

{ أَسْتَكْبَرْتَ أَمْ كُنتَ مِنَ الْعَالِينَ}

(“Were you arrogant [then], or were you [already] among the haughty?”) (Saad: 75)

Hence, Allah (swt) is the Most High because He is the creator of everything, consequently, He is the Most High over His creation.