Reminding the Children of Israel that They were preferred above the Other Nations 

Allah says; 

يَا بَنِي إِسْرَايِيلَ اذْكُرُواْ نِعْمَتِيَ الَّتِي أَنْعَمْتُ عَلَيْكُمْ وَأَنِّي فَضَّلْتُكُمْ عَلَى الْعَالَمِينَ
 
 

O Children of Israel! Remember My favor which I bestowed upon you and that I preferred you over the Alamin (nations). 

Allah reminds the Children of Israel of the favors that He granted their fathers and grandfathers, how He showed preference to them by sending them Messengers from among them and revealing Books to them, more so than any of the other previous nations.

Similarly, Allah said,

وَلَقَدِ اخْتَرْنَـهُمْ عَلَى عِلْمٍ عَلَى الْعَـلَمِينَ

And We chose them (the Children of Israel) over the Alamin, (nations) with knowledge. (44:32)

and,

وَإِذْ قَالَ مُوسَى لِقَوْمِهِ يَـقَوْمِ اذْكُرُواْ نِعْمَةَ اللَّهِ عَلَيْكُمْ إِذْ جَعَلَ فِيكُمْ أَنْبِيَأءَ وَجَعَلَكُمْ مُّلُوكاً وَءَاتَـكُمْ مَّا لَمْ يُوْتِ أَحَداً مِّن الْعَـلَمِينَ

And (remember) when Musa (Moses) said to his people:"O my people! Remember the favor of Allah to you:when He made Prophets among you, made you kings, honored you above the Alamin (nations)." (5:20)

Abu Jafar Ar-Razi reported that Ar-Rabi bin Anas said that Abu Al-Aliyah said that Allah's statement, 
وَأَنِّي فَضَّلْتُكُمْ عَلَى الْعَالَمِينَ
 (and that I preferred you over the Alamin) means,

"The kingship, Messengers and Books that were granted to them, instead of granting such to the other kingdoms that existed during their time, for every period there is a nation."

It was also reported that Mujahid, Ar-Rabi bin Anas, Qatadah and Ismail bin Abi Khalid said similarly.
The Ummah of Muhammad is Better than the Children of Israel 

This is the only way the Ayah can be understood, because this Ummah is better than theirs, as Allah said;

كُنتُمْ خَيْرَ أُمَّةٍ أُخْرِجَتْ لِلنَّاسِ تَأْمُرُونَ بِالْمَعْرُوفِ وَتَنْهَوْنَ عَنِ الْمُنْكَرِ وَتُوْمِنُونَ بِاللَّهِ وَلَوْ ءَامَنَ أَهْلُ الْكِتَـبِ لَكَانَ خَيْراً لَّهُمْ

You are the best of people ever raised up for mankind; you enjoin good and forbid evil, and you believe in Allah. And had the People of the Book (Jews and Christians) believed, it would have been better for them. (3:110)

Also, the Musnad and Sunan Collections of Hadith recorded that Muawiyah bin Haydah Al-Qushayri said that the Messenger of Allah said,

أَنْتُمْ تُوَفُّونَ سَبْعِينَ أُمَّةً أَنْتُمْ خَيْرُهَا وَأَكْرَمُهَا عَلَى الله

You (Muslims) are the seventieth nation, but you are the best and most honored of them according to Allah.

There are many Hadiths on this subject, and they will be mentioned when we discuss Allah's statement,
كُنتُمْ خَيْرَ أُمَّةٍ أُخْرِجَتْ لِلنَّاسِ
 (You are the best of peoples ever raised up for mankind), (3:110)