No, the world won't actually get destroyed at the end of the Kali Yuga. Rather, what's going to occur is that when the evil and decadence of the Kali Yuga reaches its zenith, Vishnu will have an incarnation (avatara) as Kalki the horse rider, to kill all the evil people and restore Dharma on earth, commencing a new Satya Yuga. Here is how the Srimad Bhagavatam describes it:
Lord Kalki will appear in the home of the most eminent brāhmaṇa of Śambhala village, the great soul Viṣṇuyaśā. Lord Kalki, the Lord of the universe, will mount His swift horse Devadatta and, sword in hand, travel over the earth exhibiting His eight mystic opulences and eight special qualities of Godhead. Displaying His unequaled effulgence and riding with great speed, He will kill by the millions those thieves who have dared dress as kings.
After all the impostor kings have been killed, the residents of the cities and towns will feel the breezes carrying the most sacred fragrance of the sandalwood paste and other decorations of Lord Vāsudeva, and their minds will thereby become transcendentally pure. When Lord Vāsudeva, the Supreme Personality of Godhead, appears in their hearts in His transcendental form of goodness, the remaining citizens will abundantly repopulate the earth. When the Supreme Lord has appeared on earth as Kalki, the maintainer of religion, Satya-yuga will begin, and human society will bring forth progeny in the mode of goodness. When the moon, the sun and Bṛhaspatī are together in the constellation Karkaṭa, and all three enter simultaneously into the lunar mansion Puṣyā — at that exact moment the age of Satya, or Kṛta, will begin.
So even if some people are killed, the world will still go on after the end of the Kali Yuga. The Satya Yuga, Treta Yuga, Dwapara Yuga, and Kali Yuga keep repeating in a cycle. Now one cycle of all four Yugas is called a Mahayuga or Chatur Yuga, and 1000 Mahayugas make up one Kalpa. A Kalpa constitutes just one day for Brahma the creator god. After the day is over, Brahma goes to sleep, and then the Pralaya, or night of Brahma, commences. The Pralaya, of equal length to a Kalpa, is the time period when the entire three worlds (the physical universe along with Devaloka and Asuraloka) are destroyed by fire emanating from the mouth of Vishnu's serpent Adiseshan. Here is how it's described in the Srimad Bhagavatam:
At the end of the day, under the insignificant portion of the mode of darkness, the powerful manifestation of the universe merges in the darkness of night. By the influence of eternal time, the innumerable living entities remain merged in that dissolution, and everything is silent. When the night of Brahmā ensues, all the three worlds are out of sight, and the sun and the moon are without glare, just as in the due course of an ordinary night.
The devastation takes place due to the fire emanating from the mouth of [Ananta], and thus great sages like Bhṛgu and other inhabitants of Maharloka transport themselves to Janaloka, being distressed by the warmth of the blazing fire which rages through the three worlds below. At the beginning of the devastation all the seas overflow, and hurricane winds blow very violently. Thus the waves of the seas become ferocious, and in no time at all the three worlds are full of water. The Supreme Lord [Vishnu] lies down in the water on the seat of Ananta, with His eyes closed, and the inhabitants of Janaloka offer unto the Lord their glorious prayers with folded hands.
And by the way, all this annihilation is overseen by Shiva the god of destruction, who engages in his cosmic Tandava dance in order to direct all this destruction.
After the Pralaya is over, Brahma wakes up and commences the creation of the three worlds again, and thus a new Kalpa begins. Now a Kalpa is already an incredibly long period of time, but it's just one day in the life of Brahma. Now imagine how long a hundred years in the life of Brahma! That is how long Brahma lives for, and it's called Mahakalpa. And after the Mahakalpa is over, Brahma dies, and then there is a period of even greater destruction, the Mahapralaya, which lasts as long as Mahakalpa. And then Brahma is reborn, marking the start of a new Mahakalpa. And the cycle begins again!