Praise be to Allah.
When the Prophet (peace and blessings of Allaah be upon him) came from Makkah to Madeenah, he used to face Bayt al-Maqdis when he prayed, and that remained the case for sixteen or seventeen months, as is proven in the two Saheehs (al-Bukhaari and Muslim), in the hadeeth of al-Baraa ibn Aazib (may Allaah be pleased with him and his father), who said: The Prophet (peace and blessings of Allaah be upon him) prayed towards Bayt al-Maqdis for sixteen or seventeen months, and he was hoping that the Qiblah would be towards the House (i.e., the Kabah)
Then after that Allaah commanded him to face the direction of the Kabah (the Sacred House), in the aayah (interpretation of the meaning): so turn your face in the direction of al-Masjid al-Haraam (at Makkah). And wheresoever you people are, turn your faces (in prayer) in that direction [al-Baqarah 2:144].
Before we answer the question about the wisdom behind this change, we must note the following points:
When we Muslims hear of a command from Allaah, we must accept it and submit to it, even if the wisdom behind it is not clear to us, as Allaah says (interpretation of the meaning): It is not for a believer, man or woman, when Allaah and His Messenger have decreed a matter, that they should have any option in their decision [al-Ahzaab 33:36]
Allaah, may He be glorified and exalted, does not give any command without there being great wisdom behind it even if we do not understand it as He says (interpretation of the meaning): That is the judgement of Allaah. He judges between you. And Allaah is All-Knowing, All-Wise. [al-Mumtahinah 60:10]
Allaah, may He be glorified and exalted, does not abrogate any rule except to replace it with something better or similar to it, as He says (interpretation of the meaning): Whatever Verse (revelation) do We abrogate or cause to be forgotten, We bring a better one or similar to it. Know you not that Allaah is able to do all things? [al-Baqarah 2:106]
Having understood this, we may note that the wisdom behind the changing of the Qiblah has several aspects, including:
It is a test for the true believer, because the true believer, unlike others, accepts the commands of Allaah. Allaah has spoken of this in the Quraan (interpretation of the meaning): And We made the Qiblah which you used to face, only to test those who followed the Messenger from those who would turn on their heels. Indeed it was great (heavy) except for those whom Allaah guided [al-Baqarah 2:143]
This ummah is the best of nations, as Allaah says (interpretation of the meaning): You are the best of peoples ever raised up for mankind [Aal Imraan 3:110]. Confirming the aayat about the Qiblah, Allaah says (interpretation of the meaning): Thus We have made of you (true Muslims) a Wasat (just) (and the best) nation [al-Baqarah 2:143]. Wasat conveys meanings of justice and of being chosen. So Allaah has chosen for this ummah goodness in all things and the best commands and rules, and thus He chose for them the Qiblah of Ibraaheem, upon whom be peace.
Imaam Ahmad reported in his Musnad (6/134-135) from Aaishah that the Prophet (peace and blessings of Allaah be upon him) said about the People of the Book (Jews and Christians): They do not envy us for anything as much as they envy us for Yawm al-Jumuah (Friday), to which Allaah has guided us and from which they have gone astray, and for the Qiblah to which Allaah has guided us and from which they have gone astray, and for our saying Aameen behind the imaam.(For more information on this subject, please refer to Badai al-Fawaaid by Ibn al-Qayyim, may Allaah have mercy on him, 4/157-174).
And Allaah knows best.