How do Jehovah's Witnesses respond to the charge that they are false prophets (because their end-of-the-world predictions didn't come true)?

This appears to be one of their answers:

Out of zeal and enthusiasm for the vindication of Jehovah's name, Word and purposes, and the desire for the new system, some of his servants have at times been premature in their expectations.

— Watchtower 1979 Jul 1 p.29, How Jehovah Guides His People

In their publication Reasoning from the Scriptures they address this question. They don't claim to be inspired prophets but faithful students of the scriptures who in their constant efforts to keep on the watch have made errors in regard to their expectations on end times bible prophecy. Below is the pertinent quote from that book:

Have not Jehovah’s Witnesses made errors in their teachings?

Jehovah’s Witnesses do not claim to be inspired prophets. They have made mistakes. Like the apostles of Jesus Christ, they have at times had some wrong expectations.—Luke 19:11; Acts 1:6.

The Scriptures provide time elements related to Christ’s presence, and Jehovah’s Witnesses have studied these with keen interest. (Luke 21:24; Dan. 4:10-17) Jesus also described a many-featured sign that would tie in with the fulfillment of time prophecies to identify the generation that would live to see the end of Satan’s wicked system of things. (Luke 21:7-36) Jehovah’s Witnesses have pointed to evidence in fulfillment of this sign. It is true that the Witnesses have made mistakes in their understanding of what would occur at the end of certain time periods, but they have not made the mistake of losing faith or ceasing to be watchful as to fulfillment of Jehovah’s purposes. They have continued to keep to the fore in their thinking the counsel given by Jesus: “Keep on the watch, therefore, because you do not know on what day your Lord is coming.”—Matt. 24

Additionally the definition of false prophet provided in this book is quite specific and does not therefore apply to the mistakes made by their scholars in interpreting scripture. Below is the definition:

False Prophets Definition: Individuals and organizations proclaiming messages that they attribute to a superhuman source but that do not originate with the true God and are not in harmony with his revealed will.

For more info from this book published by Jehovah's Witnesses see: False Prophets.

Additionally the Watchtower of January 2013 directly addresses this question:

HAVE JEHOVAH’S WITNESSES GIVEN INCORRECT DATES FOR THE END? Jehovah’s Witnesses have had wrong expectations about when the end would come. Like Jesus’ first-century disciples, we have sometimes looked forward to the fulfillment of prophecy ahead of God’s timetable. (Luke 19:11; Acts 1:6; 2 Thessalonians 2:1, 2) We agree with the sentiment of longtime Witness A. H. Macmillan, who said: “I learned that we should admit our mistakes and continue searching God’s Word for more enlightenment.”

Why, then, do we continue to highlight the nearness of the end? Because we take seriously Jesus’ words: “Keep looking, keep awake.” The alternative, to be found “sleeping” by Jesus, would prevent us from gaining his favor. (Mark 13:33, 36) Why?

Consider this example: A lookout in a fire tower might see what he thinks is a wisp of smoke on the horizon and sound what proves to be a false alarm. Later, though, his alertness could save lives.

Likewise, we have had some wrong expectations about the end. But we are more concerned with obeying Jesus and saving lives than with avoiding criticism. Jesus’ command to “give a thorough witness” compels us to warn others about the end.—Acts 10:42.

We believe that even more important than focusing on when the end will come, we must be confident that it will come, and we must act accordingly. We take seriously the words of Habakkuk 2:3, which says: “Even if [the end] should delay [compared to what you thought], keep in expectation of it; for it will without fail come true. It will not be late.”

Additionally there is this footnote in an Awake magazine that makes it clear that we have long been aware that we are not inspired nor infallible:

Jehovah’s Witnesses, in their eagerness for Jesus’ second coming, have suggested dates that turned out to be incorrect. Because of this, some have called them false prophets. Never in these instances, however, did they presume to originate predictions ‘in the name of Jehovah.’ Never did they say, ‘These are the words of Jehovah.’ The Watchtower, the official journal of Jehovah’s Witnesses, has said: “We have not the gift of prophecy.” (January 1883, page 425) “Nor would we have our writings reverenced or regarded as infallible.” (December 15, 1896, page 306) The Watchtower has also said that the fact that some have Jehovah’s spirit “does not mean those now serving as Jehovah’s witnesses are inspired. It does not mean that the writings in this magazine The Watchtower are inspired and infallible and without mistakes.” (May 15, 1947, page 157) “The Watchtower does not claim to be inspired in its utterances, nor is it dogmatic.” (August 15, 1950, page 263) “The brothers preparing these publications are not infallible. Their writings are not inspired as are those of Paul and the other Bible writers. (2 Tim. 3:16) And so, at times, it has been necessary, as understanding became clearer, to correct views. (Prov. 4:18)” —February 15, 1981, page 19.