These verses are part of a prophecy that was fulfilled in type in AD 69 when Christians fled just before the final Roman siege of Jerusalem began, prior to its complete destruction in AD 70. Christ knew that those with small children and nursing babies would have it especially hard. (He uses "woe" compassionately in Matthew 24:19, unlike His disparaging use of the term in Matthew 23:13.)
Perhaps we can understand why Christ said to pray that their flight from Jerusalem be not in the winter or on the Sabbath (verse 20) by noting what occurred in AD 70. According to the Jewish historian Josephus, Jerusalem was then divided into several factions. One faction wanted to surrender to the Romans and avoid destruction. Another faction, determined to fight the Romans no matter how hopeless the outcome, would not allow the rest to leave the city to surrender. The only way one could escape was to go out with hoes, rakes, and the like, under the pretense of carrying on regular duties in the fields. Under these conditions, one could not take extra clothing, other belongings, or small children without being noticed and captured.
The Sabbath and winter conditions compounded the problem. During these times, people would not be going out of the city to work in the fields. If the time to flee came then, few if any could escape. Thus, Christians were to pray that the time to flee would not fall in the winter or on the Sabbath. It should also be noted that a problem of conscience might affect some whose only opportunity to flee occurred on the Sabbath. For these people, it would be best if they pray for God not to allow that to happen.
The time is coming again, during the Great Tribulation, when Jerusalem will be surrounded by armies. Any Christians there at the time will have to flee to a place of safety to avoid the terrible bloodshed that is prophesied to occur. Christians need to pray that they are not faced with any of these conditions when the time comes.