And they were [therefore] followed in this world with a curse and [as well] on the Day of Resurrection. Unquestionably, ʿAad denied their Lord; then away with ʿAad, the people of Hūd.
(My Lord will make another people succeed you, ) This refers to a group of people who will worship Allah alone, without associating anything with Him. This also implies that the polytheists do not bother Allah and they do not harm Him in the least with their disbelief. To the contrary, their disbelief merely harms their own selves.
إِنَّ رَبِّى عَلَى كُلِّ شَىْءٍ حَفِيظٌ
(Surely, my Lord is Guardian over all things.) This means that Allah is a Witness and Guardian over the statements of His servants and their actions. He will give them due recompense for their actions. If they do good deeds, He will reward them with good. If they do evil, He will punish them with evil.
After that, the next eight verses (61-68) carry the story of Sayyidna Salih peace be upon him. He was sent to the people of Thamud, another branch of the tribe of ` Ad. He too invited his people to believe in Tauhid, the Oneness of Allah. As customary, his people belied him and insisted that he must first make a she-camel of a specified description come out from the rock of the mountain in front of them in order to have them believe that he was a true prophet of Allah.
Sayyidna Salih (علیہ السلام) warned them: If Allah Ta` ala shows you your custom-ordered miracles, and you still fail to believe, then, according to the customary practice of Allah, the punishment will come and destroy all of you. But, they remained obstinate and Allah Ta` a1a, in His perfect power, made the miracle they demanded manifest before them. The rock of the mountain in front of them split asunder. Out from it came the she-camel bearing the features specified by them. Allah Ta` ala ordered them not to harm that she-camel in any manner, otherwise the punishment will come upon them. But, they disobeyed this order as well, and killed the she-camel. Consequently, they were seized by Allah Ta` ala, Sayyidna Salih (علیہ السلام) and his companions were saved from the punishment, and the rest of those people were destroyed by a Sound which was severe and terrifying.
Tazkirul Quran
Maulana Wahiduddin KhanOn ayahs 60–62
Those who ignore God’s word, are ignored by God. This happens to a lesser degree in the present world but, on Doomsday, this will be a total reality. At that time all the arrogant and insolent people will be denied God’s grace; the grace of God will be shared only by those who lived in the world as His loyal and obedient subjects. God has introduced the principle of succession (istikhlaf) in this world, that is, the removal of a community from its position of prominence and the installing of another community in its place. This temporary prominence in the world is for the purpose of putting people to the test. In the Hereafter, in the perfect world of God, such prominence will be granted only to people of true Faith as an everlasting reward. The system of this world of trial has been formulated in such a way that, here, a man is always placed between good and bad. He has the option to follow either of these two ways. Furthermore, evil is largely dominant in this world. Only the strength of theoretical arguments is on the side of goodness. On the contrary, material power weighs in on the side of evil, and that too in such great measure that its flag-bearers develop attitudes of arrogance and hauteur and create such an atmosphere of pressure in the environment that the common man loses the courage to proceed towards the Truth.
Uthmani text of the Madinah mushaf (King Fahd Complex). Translation: Saheeh International. Tafsir via Quran.com.