In the previous Surah (Surah Humaza) Allaah mentioned the condition of the Hamaaz (the one who slanders by the tongue) and the Lamaaz (the one who slanders by way of action e.g. through facial expressions); and the one who gathers wealth. Allaah mentioned how that will cause disruption.
The relevance of this Surah (Surah Fil) to the previous Surah (Surah Humaza) is that the previous Surah speaks about wealth and it causing destruction, and this Surah brings the story of those who were powerful by way of their wealth.
أَلَمۡ تَرَ كَيۡفَ فَعَلَ رَبُّكَ بِأَصۡحَـٰبِ ٱلۡفِيلِ (١)
Have you not seen how your Lord dealt with the Owners of the Elephant? (1)
This ayah is referring to people who came from Yemen to try and destroy the ka’bah. They came with a large elephant; it is said its name was Mahmood. The elephant that was commanded to charge the Ka’bah refused, but when it was turned back to Yemen it would trod.
أَلَمۡ يَجۡعَلۡ كَيۡدَهُمۡ فِى تَضۡلِيلٍ۬
Did He not make their plot go astray? (2)
This evil plot they had was turned on them and they were the ones destroyed.
وَأَرۡسَلَ عَلَيۡہِمۡ طَيۡرًا أَبَابِيلَ (٣)
And He sent against them birds, in flocks, (3)
) تَرۡمِيهِم بِحِجَارَةٍ۬ مِّن سِجِّيلٍ۬
Striking them with stones of Sijjîl (baked clay). (4)
فَجَعَلَهُمۡ كَعَصۡفٍ۬ مَّأۡڪُولِۭ (٥)
And He made them like (an empty field of) stalks (of which the corn has been eaten up by cattle). (5)
This Surah was an encouragement to the Prophet (Sallallaahu Alayhi Wa Sallam) how he will be protected from the Quraysh and those enemies around him, and how Allaah will protect that which is sacred (i.e. the house of Allaah). This Surah was also a reminder of the power of Allaah, how he controls the birds, the elephants, and everything created.
Surah Quraysh- This Surah was named as such due to Allaah protecting the people of Quraysh. The relevance of this surah to the previous one is that both Surahs mention the Quraysh. The first surah mentions the enemies who came to destroy the house, and the second Surah speaks about the unity of the people of Quraysh and how Allaah brought them together.
لِإِيلَـٰفِ قُرَيۡشٍ (١)
(It is a great Grace and protection from Allâh), for the taming of the Quraish, (1)
إِیلَـٰفِهِمۡ رِحۡلَةَ ٱلشِّتَآءِ وَٱلصَّيۡفِ (٢)
(And with all those Allâh’s Grace and Protections for their taming, We cause) the (Quraish) caravans to set forth safe in winter (to the south), and in summer (to the north without any fear), (2)
In the winter they would travel to Yemen and in the summer they would travel to Syria. Due to this safety, they were able to enjoy specific things in the winter and summer.
فَلۡيَعۡبُدُواْ رَبَّ هَـٰذَا ٱلۡبَيۡتِ (٣)
So let them worship (Allâh) the Lord of this House (the Ka’bah in Makkah). (3)
ٱلَّذِىٓ أَطۡعَمَهُم مِّن جُوعٍ۬ وَءَامَنَهُم مِّنۡ خَوۡفِۭ (٤)
(He) Who has fed them against hunger, and has made them safe from fear. (4)