You have asked a very good question.
First, God has His people whom He elected. This is clear in the Bible I and assume that that is already accepted as truth.1Pe 2:9 But ye are a elect race, a royal priesthood, a holy nation, a people for <I>God's</I> own possession, that ye may show forth the excellencies of him who called you out of darkness into his marvellous light: Eph 1:4 even as he chose us in him before the foundation of the world, that we should be holy and without blemish before him in love:Mat 22:14 For many are called, but few chosen.
Rom 8:29 For whom he foreknew, he also foreordained <I>to be</I> conformed to the image of his Son, that he might be the firstborn among many brethren: Rom 8:30 and whom he foreordained, them he also called: and whom he called, them he also justified: and whom he justified, them he also glorified. Secondly, these people, the elect, will come unto salvation.
Joh 6:39 And this is the will of him that sent me, that of all that which he hath given me I should lose nothing, but should raise it up at the last day. Joh 6:40 For this is the will of my Father, that every one that beholdeth the Son, and believeth on him, should have eternal life; and I will raise him up at the last day.
These two verses are very similar. They both pertain to the Father's will and they both say a certain group of people will be raised up at the last day. The point is, the two groups of people, those whom the father has given to Christ (the elect) and those who behold and believe on Christ are the very same people. Meaning the elect will believe in Christ and they will have eternal life.
How then do the elect become aware of God's grace towards them? They aren't born with this innate understanding that they are chosen by God. When and how then do they become aware?
What should be asked is what initiates salvation? Does man or God? Also, if man is the initiator instead of God then you can still ask the same question, "How does a Christian know if they are a Christian?"
Joh 6:44 No man can come to me, except the Father that sent me draw him: and I will raise him up in the last day.
We know that no man can come unto Christ unless they are drawn by His Father. Thusly, those who do come to Christ where certainly drawn by the Father. Also, those who are drawn by the Father WILL be raised up in the last day.
Now, where in all this do we become aware of our new condition. Here is a passage that shows this change. We must first understand man's condition before the law. The law brings sin and death since we are unable to fulfill it; this is called sin. However, carnal man is a slave to such sin and is unable to break free. Rom 7:14 For we know that the law is spiritual: but I am carnal, sold under sin. However, Christ does this for us. He fullfilled the law and through His attoning death on the cross his elect are counted as righteous; those who have vicariously fullfilled the law. In this, we are made aware of our new condition where we see our need for a saviour in the fact that we cannot fullfill the law without Christ. Here is the passage:
Rom 7:22 For I delight in the law of God after the inward man:
Rom 7:23 but I see a different law in my members, warring against the law of my mind, and bringing me into captivity under the law of sin which is in my members.
Rom 7:24 Wretched man that I am! who shall deliver me out of the body of this death? Rom 7:25 I thank God through Jesus Christ our Lord. So then I of myself with the mind, indeed, serve the law of God; but with the flesh the law of sin.
Verse 24 shows our initial condition and our awareness of our helplessness. Only God can can deliver us from ourselves, but when He does we have an experience wherein we realize our old condition and how in it were were helpless. This makes us aware of exactly why and what Christ did for us. When Christ delivers us we are able to serve the law of God with our minds.Luk 24:45 Then opened he their mind, that they might understand the scriptures;
Not only our mind is changed but our heart is also.
Eze 36:26 A new heart also will I give you, and a new spirit will I put within you; and I will take away the stony heart out of your flesh, and I will give you a heart of flesh.
You also asked, "where does our assurance come from?"1Th 1:4 knowing, brethren beloved of God, your election, 1Th 1:5 how that our gospel came not unto you in word only, but also in power, and in the Holy Spirit, and <I>in</I> much assurance
You see, if God is the one whom initiated our salvation then it is also up to God to keep us in salvation. As stated earlier God will not lose any of his elect (John 6:39).
What better assurance is there than to know that salvation is in God's hands from start to finish, rather than in ours.
Phi 2:13 for it is God who worketh in you both to will and to work, for his good pleasure.Phi 1:6 being confident of this very thing, that he who began a good work in you will perfect it until the day of Jesus Christ:
God Bless!