Here is the text of the verse in question:
Isaiah 65:20 - "There shall be no more thence an infant of days, nor an old man that hath not filled his days: for the child shall die an hundred years old; but the sinner being an hundred years old shall be accursed."
First, we must understand the context of the verse so that we can know how to apply it. We get this by looking at the verses close to this verse in Isaiah 65. Verse 19 speaks of a time of rejoicing in Jerusalem and among God's people. Verses 21-23 teach that it will be a time when labor will not be spent in vain. Those who build the houses will get to live in them. Those who plant the vineyards will get to eat of the vineyard. Verse 24 says that it is a time when God will answer before they even call and while they are yet speaking.
As far as I am concerned, verse 25 settles the time in question. It says that the wolf and the lamb shall feed together and the lion shall eat straw like an ox. This certainly refers to the future kingdom age when Christ comes down to earth a second time to rule and reign among men. It is the time doctrinally called the millennium. This will be a time of special blessing on the earth and it is the time described in verse 20. Now, let us look back to that verse phrase by phrase.
"There shall be no more thence an infant of days"
This phrase may seem difficult without the rest of the verse. But I believe it refers to the absence of death among babies. One of the saddest things we face in this world is the death of a baby. It seems so wrong to bring the child into the world only to die after a short time. During the kingdom age, this will not happen. There will be no baby funerals.
"nor an old man that hath not filled his days"
There will also be no such thing as a man who has grown old before he has had time to fulfill his purpose in life. Presently, life goes by so fast. About the time we begin to learn to live, we find that our life is about over with. When Christ rules on earth, this will not be the case. Life spans will be much longer than they are today much like they were before the flood.
"for the child shall die an hundred years old"
In fact, life spans will be so long that anyone who dies at the age of one hundred will be considered a child. This is just a way to emphasize the length of life. The world will be full of Methuselah's. To die at one hundred is to die as a child.
"but the sinner being an hundred years old shall be accursed."
The exception is found in the man or woman who rebels against God (yes, this will be possible during the kingdom age). This man will show his age at one hundred; his body will be under a curse. Zechariah 14 refers to the kingdom age as well. During this time, those nations that refuse to come up to Jerusalem for the feast of tabernacles will find that they have no rain the next year (Zechariah 14:16-19). This is one of the ways in which the Son will rule the earth with a rod of iron (Psalm 2:7-9). In Isaiah 65:20, we find that sin will tell on the body in such a way that a righteous man is a child at one hundred but a sinner is accursed at the same age.