Matthew 27:51-53 tells us what happened right after Jesus Christ died: "Then, behold, the veil of the temple was torn in two from top to bottom; and the earth quaked, and the rocks were split, and the graves were opened; and many bodies of the saints who had fallen asleep were raised; and coming out of the graves after His resurrection, they went into the holy city and appeared to many."
This resurrection of several believers was one of the phenomena accompanying the resurrection of Jesus Christ to underscore that monumental event. These saints came back to physical life. (In the Bible, the word saints means those who are sanctified or set apart as holy, meaning all of God's true followers.) We know from such scriptures as 1 Thessalonians 4:15-17 and 1 Corinthians 15:51-52 that God has resurrected no one to eternal life as an immortal spirit yet—except Christ.
For example, 1 Thessalonians 4:15-17 says: "For this we say to you by the word of the Lord, that we who are alive and remain until the coming of the Lord will by no means precede those who are asleep. For the Lord Himself will descend from heaven with a shout, with the voice of an archangel, and with the trumpet of God. And the dead in Christ will rise first. Then we who are alive and remain shall be caught up together with them in the clouds to meet the Lord in the air. And thus we shall always be with the Lord."
These references teach us that God will change the righteous dead along with living Christians to immortal beings when Jesus Christ returns.
After God brought the people mentioned in Matthew 27:52-53 back to life, they undoubtedly went back to their homes, where many acquaintances saw them. It's hard to imagine how utterly startling such an experience would be for their relatives and friends!
The Bible says nothing further about these people God resurrected at Christ's death, leaving us to conclude that they eventually died (again) and their families buried them (again). Along with all other saints who died, they await in their graves their resurrection to spirit life.