Surah al-Nasr

Shaane Nuzool (Reason for revelation)
It has been mentioned in the books of Tafseer that this Surah was one of the last Surahs to be revealed completely. Ibn Umar RA narrates that this Surah was revealed during the Farewell Pilgrimage. Ibn Abbas RA mentioned that when this Surah was revealed, the Messenger SAW had been informed of his passing and that His time to leave the world had arrived. [Musnad Ahmad, Ibn Jarrir, Ibn Mundhir, Ibn Marduyah]

Lessons to be learnt from this Surah
The first message to be conveyed in this Surah is the victory of Islam. The majority of Ulama (scholars) say that 'Fatah' refers to the conquest of Makkah. It also refers to the arrival of many ambassadors from various other regions who came in order to spend time in the company of the Messenger SAW. They had heard about the message of the Prophet SAW and wanted to accept Islam. They would take their Shahaadah at the hands of the Messenger SAW and would return and then their entire tribe would accept Islam. In the space of a decade, people had gone from mocking  the message of Islam to arriving from far and wide to accept it en masse. 

The Conquest of Makkah was the first victory which did not occur through bloodshed or violence. The Muslims were so humble in their success that the only 'celebration' they carried out was words of Praise to the Almighty, thanking Him for His Assistance. 

"And if you are Grateful, then I will increase you in Bounties"
Surah Ibrahim: 7

وَرَأيْتَ اْلنّاَسَ means 'And You will see'. This phrase is directed to the Messenger SAW and refers to the gift of being able to see people gathering in large numbers to accept Islam. Before this, He SAW would get very worried that people may not accept Islam. 

The second message of this Surah is the instructions of Tasbih (Remembrance of Allah), Hamd (Praise of Allah) and Istighfaar (Seeking forgiveness). After the Conquest of Makkah, the Muslims purified the Ka'bah (House of Allah) of the idols and so Allah SWT instructs the Believers to purify their hearts through His Remembrance and Praise. 
It is a reminder that if one does not thank Allah for the blessings He bestows upon us, be it a victory over our enemies, or for the sustenance He provides us with, then Allah can easily take these Ne'mat (blessings) away. Not showing appreciation can lead to arrogance developing in the heart and to counter this a person should remember Allah often through seeking forgiveness, asking for His assistance and praising Him. 

May the Almighty Allah give us the ability to praise Him, turn towards Him for help and keep us steadfast on His Deen. Ameen.