In the Name of Allaah, the Most Merciful, the ever Merciful.
What is the meaning of the supplication that the Prophet, sallallaahu 'alayhi wa sallam (may Allaah exalt his mention), encouraged us to say after the Athaan (call for prayer)?
May Allaah reward you.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

It was narrated on the authority of Jaabir that the Prophet said: "Whoever says upon hearing the Athaan (call for prayer): 'Allahumma Rabba hathihi-d-da‘ watit-taammati was-salatil qaa’imah, aati Muhammadan al-waseelata wal-fadheelah, wab‘athhu maqaaman mahmoodan-il-lathi wa‘adtah (O Allaah, Lord of this perfect call (call for monotheism) and of the current prayer (which is about to start) please give Muhammad Al-Waseelah (the highest rank in Paradise) and a superior status over all creatures and raise him to the station of praise and glory, (that is the highest degree in Paradise), which You promised him,' then he will be granted my intercession on the Day of Resurrection)."

The meaning of Al-Waseelah, the highest rank in paradise, was mentioned in another Hadeeth. In this Hadeeth the Prophet encouraged us after hearing the call for prayer to beg Allaah The Almighty to grant him this rank, since it is a rank in Paradise fitting only for one of the servants of Allaah The Almighty. The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, also said that anyone who asks that he (the Prophet, sallallaahu ‘alayhi wa sallam,) be given Al-Waseelah will be assured of the intercession of the Prophet sallallaahu ‘alayhi wa sallam on his behalf on the Day of Judgment.

The promise to raise the Prophet to the Maqaam Mahmood, for which people will thank Allaah on the Day of Resurrection, is mentioned in the verse (which means): {It may be that your Lord will raise you to Maqam Mahmood (a station of praise and glory, i.e. the highest degree in Paradise!)}  [Quran 17:79]

Allaah Knows best.