Praise be to Allah.
The meaning of tasbeeh (glorifying Allah) is declaring Allah, may He be exalted, to be far above any shortcoming or fault. So when you say “Subhaan Allah (Glory be to Allah)”, what it means is: I declare You, O Lord, to be far above and free of any shortcoming or fault.
We have previously explained the meaning of tasbeeh in detail in the answer to question no. 170072.
The meaning of al-‘Azeem (the Most Great) is the One Who is possessed of ultimate greatness. The meaning of al-A‘la (the Most High) is the One Who is exalted in His Essence, and is exalted in His attributes.
What is required of the worshipper is to reflect upon the meaning of what he recites of Qur’an and adhkaar. In his recitation of Qur’an and dhikr he is required to bear in mind the meaning connected to these words, because every word that he utters in prayer has subtle wisdom and meanings, of which he will attain the goodness and benefits commensurate with his level of focus of mind and reflection on the meaning, and he will lose out on those benefits commensurate with the extent to which he is not able to focus and reflect on what he is reciting.
Bearing the meanings in mind is not limited only to the words of the prayer; rather it is also required to bear in mind the great significance of the actions of the prayer.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: One of the means that will help the worshipper not to let his mind wander whilst he is praying is to follow what he is saying and doing, and reflect upon the great wisdom behind the prescription of these words and actions. In the case of rukoo‘ (bowing), for example, it is prescribed for the worshipper to glorify his Lord in his words and his actions. Hence the Prophet (blessings and peace of Allah be upon him) said: “…and in rukoo‘, glorify the Lord.” Bowing before Allah is glorifying Him by one’s actions, and saying “Subhaana Rabbiy al-‘Azeem (Glory be to my Lord the Most Great)” is glorifying Him by one’s words. All that is left is for the worshipper to glorify Him in his heart, and this can only be attained with presence of mind. Hence in rukoo‘, there is glorification of Allah in one’s words and actions, and also in one’s heart.
End quote from Fataawa Noor ‘ala ad-Darb (8/2).
Ibn Rajab said: When the worshipper shows humility to his Lord by bowing and prostrating, he is attributing to his Lord the attributes of might, majesty, greatness and exaltedness. It is as if he is saying: Humility and humbleness are my attributes, and exaltedness, greatness and majesty are Your attributes. Hence it is prescribed for the worshipper when bowing to say: “Subhaan Rabbiy al-‘Azeem (Glory be to my Lord the Most Great)” and when prostrating to say: “Subhaan Rabbiy al-A‘laa (Glory be to my Lord Most High).” The Prophet (blessings and peace of Allah be upon him) sometimes used to say when bowing and prostrating: “Subhaana Dhi’l-jabarooti wa’l-malakooti wa’l-kibriyaa’ wa’l-‘azamah (Glory be to Him Who possesses might, sovereignty, majesty and greatness).”
End quote from al-Khushoo‘ fi’s-Salaah (p. 41-43).
The questioner says: Is it obligatory to reflect upon the meaning when repeating these phrases, or is it that what is required is to reflect upon the majesty and perfection of Allah when bowing, and His exaltedness?
The answer is that what is required of the worshipper is to reflect upon the meaning of what he is saying of Qur’an and dhikr, and on what he is doing of actions, for bowing and prostration are prescribed for the glorification of Allah, may He be exalted, and the prescribed dhikr [in prayer] has the meaning of glorification and veneration.
Based on that, whoever reflects on the meanings of what he says of adhkaar when bowing and prostrating will inevitably lead him to reflect upon the might and majesty of Allah, may He be exalted. But his reflection should be commensurate with what he is saying and doing in the prayer, and he should not overdo it, for that is waswaas (whispers) from the Shaytaan, aimed at distracting him from his prayer.
Al-Ghazaali said in Ihyaa’ ‘Uloom ad-Deen (1/150): You should understand that one of the tricks [of the Shaytaan] is to distract you in your prayer by making you remember the hereafter and think of what good deeds you could do, so as to prevent you from focusing on what you are reciting. Therefore you should realise that anything that distracts you from focusing on the meanings of what you are reciting comes under the heading of waswaas. End quote.
And Allah knows best.