Allah's statement:

﴿وَلِتُكَبِّرُواْ اللَّهَ عَلَى مَا هَدَاكُمْ﴾

(...and that you must magnify Allah ﴿i.e., to say Takbir (Allahu Akbar: Allah is the Most Great)﴾ for having guided you) means: So that you remember Allah upon finishing the act of worship. This is similar to Allah's statement:

﴿فَإِذَا قَضَيْتُم مَّنَـسِكَكُمْ فَاذْكُرُواْ اللَّهَ كَذِكْرِكُمْ ءَابَآءَكُمْ أَوْ أَشَدَّ ذِكْرًا﴾

(So when you have accomplished your Manasik, (rituals) remember Allah as you remember your forefathers or with far more remembrance.) (2:200) and:

﴿فَإِذَا قُضِيَتِ الصَّلَوةُ فَانتَشِرُواْ فِى الاٌّرْضِ وَابْتَغُواْ مِن فَضْلِ اللَّهِ وَاذْكُرُواْ اللَّهَ كَثِيراً لَّعَلَّكُمْ تُفْلِحُونَ ﴾

(...Then when the (Jumu`ah) Salah (prayer) is ended, you may disperse through the land, and seek the bounty of Allah (by working), and remember Allah much, that you may be successful.) (62:10) and:

﴿فَاصْبِرْ عَلَى مَا يَقُولُونَ وَسَبِّحْ بِحَمْدِ رَبِّكَ قَبْلَ طُلُوعِ الشَّمْسِ وَقَبْلَ الْغُرُوبِ - وَمِنَ الَّيْلِ فَسَبِّحْهُ وَأَدْبَـرَ السُّجُودِ ﴾

(...and glorify the praises of your Lord, before the rising of the sun and before (its) setting. And during a part of the night, glorify His praises, and after the prayers.) (50:39, 40)

This is why the Sunnah encouraged Tasbih (saying Subhan Allah, i.e., all praise is due to Allah), Tahmid (saying Al-Hamdu Lillah, i.e., all the thanks are due to Allah) and Takbir (saying Allahu Akbar, i.e., Allah is the Most Great) after the compulsory prayers. Ibn `Abbas said, "We used to know that Allah's Messenger has finished the prayer by the Takbir.'' Similarly, several scholars have stated that reciting Takbir the during `Id-ul-Fitr was specified by the Ayah that states:

﴿وَلِتُكْمِلُواْ الْعِدَّةَ وَلِتُكَبِّرُواْ اللَّهَ عَلَى مَا هَدَاكُمْ﴾

((He wants that you) must complete the same number (of days), and that you must magnify Allah ﴿i.e., to say Takbir (Allahu Akbar: Allah is the Most Great)﴾ for having guided you...) Allah's statement:

﴿وَلَعَلَّكُمْ تَشْكُرُونَ﴾

(...so that you may be grateful to Him.) means: If you adhere to what Allah commanded you, obeying Him by performing the obligations, abandoning the prohibitions and abiding by the set limits, then perhaps you will be among the grateful.

﴿وَإِذَا سَأَلَكَ عِبَادِي عَنِّي فَإِنِّي قَرِيبٌ أُجِيبُ دَعْوَةَ الدَّاعِ إِذَا دَعَانِ فَلْيَسْتَجِيبُواْ لِى وَلْيُؤْمِنُواْ بِى لَعَلَّهُمْ يَرْشُدُونَ ﴾

(186. And when My servants ask you (O Muhammad concerning Me, then answer them), I am indeed near (to them by My knowledge). I respond to the invocations of the supplicant when he calls on Me (without any mediator or intercessor). So let them obey Me and believe in Me, so that they may be led aright.)