When non-christians ask us how we know that what we believe in is the truth, and how do we prove it, how do we answer that question?

First of all, I want to assure you that we are commanded to answer the questions of those who ask us why we believe what we believe. 1 Peter 3:15But sanctify the Lord God in your hearts: and be ready always to give an answer to every man that asketh you a reason of the hope that is in you with meekness and fear:
See All... states, "But sanctify the Lord God in your hearts: and be ready always to give an answer to every man that asketh you a reason of the hope that is in you with meekness and fear." We are to be ready to answer the questions we receive about our faith. In fact, there is an entire field of Bible studies meant to answer these questions. It is called the field of apologetics and it deals with answering the objections to Christian faith and doctrine.

However, that does not mean that we have to have all knowledge and be quick to answer any objection to Christianity. No one automatically has the answers to every question. One of the best ways to learn to answer these questions is to seek an answer when someone challenges you with a question you had not considered before. Tell them that you do not have a good answer right now but that you will seek to find one for them. And then do just that. You can seek the advice and counsel of others, search the scriptures, and pray for God to show you how to answer their question. The entire process is a way that you can grow in the things of the Lord.

The question you have been asked is a complex one and could be drawn out easily to book length. However, I am sure you want something simpler than that. You may find another approach that is helpful to you, but I will give you an approach I might take if asked this question.

I might initially respond to their question with this question: "What is your authority for truth?" If they did not understand my meaning (as is likely the case), I would explain to them that the world is full of conflicting authorities that all claim to be telling the truth. In order to determine what is true in any given case, each person has to decide who or what he will believe. When two authorities conflict, the one that I accept over the other is my authority for truth. The authority I take above all others is my final authority for truth. Here are some of the authorities people have for truth with scriptures that describe them:

1. Tradition: the authority of man’s long-time practice (Matthew 15:1-9 [1] Then came to Jesus scribes and Pharisees, which were of Jerusalem, saying, [2] Why do thy disciples transgress the tradition of the elders? for they wash not their hands when they eat bread. [3] But he answered and said unto them, Why do ye also transgress the commandment of God by your tradition? [4] For God commanded, saying, Honour thy father and mother: and, He that curseth father or mother, let him die the death. [5] But ye say, Whosoever shall say to his father or his mother, It is a gift, by whatsoever thou mightest be profited by me; [6] And honour not his father or his mother, he shall be free. Thus have ye made the commandment of God of none effect by your tradition. [7] Ye hypocrites, well did Esaias prophesy of you, saying, [8] This people draweth nigh unto me with their mouth, and honoureth me with their lips; but their heart is far from me. [9] But in vain they do worship me, teaching for doctrines the commandments of men.
   See All..., esp.v.3,6,9; Colossians 2:8Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ.
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2. Philosophy: the authority of man’s greatest wisdom (Colossians 2:8Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ.
   See All...; Acts 17:18Then certain philosophers of the Epicureans, and of the Stoicks, encountered him. And some said, What will this babbler say? other some, He seemeth to be a setter forth of strange gods: because he preached unto them Jesus, and the resurrection.
   See All..., with v.16,22,23)
3. Science: the authority of observable facts and repeatable physical laws (1 Timothy 6:20O Timothy, keep that which is committed to thy trust, avoiding profane and vain babblings, and oppositions of science falsely so called:
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4. Scholarship: the authority of academic learning and agreed upon knowledge (2 Timothy 3:7Ever learning, and never able to come to the knowledge of the truth.
   See All...; Isaiah 29:9-12 [9] Stay yourselves, and wonder; cry ye out, and cry: they are drunken, but not with wine; they stagger, but not with strong drink. [10] For the LORD hath poured out upon you the spirit of deep sleep, and hath closed your eyes: the prophets and your rulers, the seers hath he covered. [11] And the vision of all is become unto you as the words of a book that is sealed, which men deliver to one that is learned, saying, Read this, I pray thee: and he saith, I cannot; for it is sealed: [12] And the book is delivered to him that is not learned, saying, Read this, I pray thee: and he saith, I am not learned.
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5. Pragmatism: the authority of what brings the desired results (1 Corinthians 1:18-25 [18] For the preaching of the cross is to them that perish foolishness; but unto us which are saved it is the power of God. [19] For it is written, I will destroy the wisdom of the wise, and will bring to nothing the understanding of the prudent. [20] Where is the wise? where is the scribe? where is the disputer of this world? hath not God made foolish the wisdom of this world? [21] For after that in the wisdom of God the world by wisdom knew not God, it pleased God by the foolishness of preaching to save them that believe. [22] For the Jews require a sign, and the Greeks seek after wisdom: [23] But we preach Christ crucified, unto the Jews a stumblingblock, and unto the Greeks foolishness; [24] But unto them which are called, both Jews and Greeks, Christ the power of God, and the wisdom of God. [25] Because the foolishness of God is wiser than men; and the weakness of God is stronger than men.
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6. Experience: the authority of personal experience (Romans 10:2-3 [2] For I bear them record that they have a zeal of God, but not according to knowledge. [3] For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God.
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I would challenge the person asking the question to figure out what their authority for truth is. Then, I would tell them that my authority for truth is the teaching of the word of God--the Bible. I would declare the word of God as the only God-given authority for truth (John 8:31-32 [31] Then said Jesus to those Jews which believed on him, If ye continue in my word, then are ye my disciples indeed; [32] And ye shall know the truth, and the truth shall make you free.
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See All...). I would explain that there are many evidences supporting the authority of God's word. These evidences include fulfilled prophecy, influence of the Bible in individuals and in world history, scientifically accurate statements in scripture, the endurance of the Bible through the ages, and others. If the person wanted to go further, it would open the door to have Bible studies with them and see what God would do in their heart. If they reject it outright, then I have still given them an answer.

Also, I would push them to define their final authority for truth. Then, I would point out that their authority for truth has flaws as well. Science and philosophy are continually changing. Experience differs from one person to another. The others have their flaws too. This way, we answer their questions by both defending what we believe and challenging their beliefs. Learning to answer objections to the faith is a lifelong process, but perhaps this will be a good start.