Very good and deep question indeed!
I will mention some qualities of this verse and also, mention the significance of this in light of those qualities, Insha Allah.
A point of benefit from this Ayah is that reciting it in the morning and evening is a major cause for one to be protected by Allah. This surah is one of the Mu`awwi- dhatayn (Surah Al Falaq & An-Naas)
It was narrated by ‘Abdullah bin Khubayb from his father from the Messenger of Allah (salallahu alahyhi wassalam) who said:
“Recite!” I said, "Messenger of Allah, what should I recite? The
Messenger of Allah (salallahu alayhi wassalam) said, "Recite Qul huwa
Allahu Ahad (Soorat-ul-Ikhlaas) and the two chapters of seeking refuge
(Soorat-ul-Falaq and Soorat-un-Naas) when you enter upon evening and
when you enter upon the morning three times each and it will suffice
you from every thing.”
It was related by Aboo Daawud (#5082) with its wording and by at-Tirmidhee (#3575) and al-‘Alaamah Ibn Baaz (may Allah have mercy upon him) graded it as being Hasan in Tuhfat-ul-Akhyaar.
This is called one of the surahs of Isti`aadah (seeking refuge) [Among the others mentioned in the hadith]. And in this surah, the name of “ar-Rabb” and the Ruboobiyyah (Lordship of Allah) mentioned therein has been tied to the Daybreak (al-falaq). Similarly in Soorat-un-Naas, “ar-Rabb”, “al-
Malik” has been tied to Mankind (an-naas). This affirms the power of Allah and his lordship over day and night and the mankind.
It is important to remember that The Night and Day are Signs of the Great Power of Allah.
Coming to the verse, Ibn Abi Hatim recorded that Jabir said, "Al-Falaq is the morning." Al-'Awfi reported from Ibn `Abbas, "Al-Falaq is the morning." The same has been reported from Mujahid, Sa'id bin Jubayr, Abdullah bin Muhammad bin 'Aqil, Al-Hasan, Qatadah, Muhammad bin Ka'b Al-Qurazi and Ibn Zayd. Malik also reported a similar statement from Zayd bin Aslam.
Later, in the Ayah, He mentions evil of darkness of night.
And from the evil of the Ghasiq when Waqab (Surat Al-Falaq :3)
Mujahid said, "Ghasiq is the night, and `when it Waqab' refers to the setting of the sun.'' Al-Bukhari mentioned this from him. Ibn Abi Najih also reported a similar narration from him (Mujahid). Lot of other early scholars said the same.
Ibn Zayd said, "The Arabs used to say, `Al-Ghasiq is the declination (of the position) of the heavenly body known as Pleiades. The number of those who were ill and stricken with plague would increase whenever it would decline, and their number would lessen whenever it rose."
Ibn Jarir said, "Others have said that it is the moon."
The support for the people who hold this position (that it means the moon) is a narration that Imam Ahmad recorded from Al-Harith bin Abi Salamah. He said that `A'ishah said,
"The Messenger of Allah took me by my hand and showed me the moon
when it rose, and he said, "Seek refuge with Allah from the evil of
this Ghasiq when it becomes dark.''
At-Tirmidhi and An-Nasa'i both recorded this Hadith in their Books of Tafsir in their Sunans.
The important message
Anyway, the core of this ayah is that Allah shows you by his Signs that he has the power to remove the evil (of darkness of night) and the daybreak is sign of his power and his ability to remove the evil. That is why, the refuge must be sought only with Allah since he has the power to remove afflictions and if evil overcomes you, His refuge must be sought.
On a related note
Surat Al-'An`ām :96 also mentions Him as the cleaver of daybreak and then immediately He says, the opposite, that he has made the night for rest. Allah reminds of His ability to create diversified things in opposites. The sun, moon, and the night and day are all the signs of His power.
Source: Tafsir and Recognizing Allah Through Some of His Ayat