Hi Doris, great question(s).
First just to clarify, I think you are asking two separate questions. One is an ontological question “what does it mean to be without sin?”, which relates to what sin is, and how one can or cannot be “in sin.” Anytime we ask a question about what something is, we are asking a question about ontology. Second, is your question “How can you know Jesus is without sin?” which is more of an epistemological question. Anytime we ask a “How can you know” question, we are dealing with epistemology.
That said, we have the “what is sin?” question and the “how do we know Jesus didn’t do it?” question. The first we derive right from Scripture, and Scripture tells us in various places and various ways that sin is something like a broken relationship with the one true God, whereby we exchange the primacy of God’s will for the primacy of our own will (or perhaps the will of some other being that is not God, e.g. Satan). So, to be with sin is to have a will that is not in sync with the will of God, but that is either apathetic or antagonistic to God’s will. To be either apathetic toward the will of God, or antagonistic toward the will of God, is a fundamental sign of one’s lack of love of God (John 14:21).
So, if sin is having a will that is either apathetic to, or antagonistic to the will of God, then the opposite of sin, call it holiness, or righteousness, is being in agreement with and acting according to the will of God, and that out of the right heart, or desire (Ezk 36:26). If being in agreement with and acting in accordance to the will of God out of the right heart is being righteous, then doing so perfectly is being “without sin.” As such, Jesus being without sin would mean that Jesus never disagreed with, nor acted contrary to the Father’s will, and never acted out of the wrong motivation, i.e. never acted out of a motivation other than love of the Father.
So, if man in his fallen nature was incapable of being perfect in this sense, then something like the “hypostatic union” the taking on of a human nature by God, would be logically necessary for a human person to ever perfectly agree with and act accordingly to the will of God. Thus, Jesus, in being both fully God and fully man, was without sin. And this is also possible if it is the case that sin is not part of what it means to be human essentially. In other words, “sinfulness” is an accidental property of mankind, not an essential one. Thus, we can be fully human and also be without sin. In fact, we will be fully human in the new creation and without sin; so we actually have this promise of being fully human and fully righteous human persons.
With regard to your second question, the epistemic one of “how we know” Jesus was like this, that will depend on our view of Scripture, especially our view of the doctrine of Revelation and Inspiration. If we have good evidence to believe the Bible is the revealed Word of God, and inspired and even inerrant in all it teaches, then, in combination with the Holy Spirit’s work, we can take at face value the words of the author of Hebrews who says:
14 Since then we have a great high priest who has passed through the heavens, Jesus, the Son of God, let us hold fast our confession. 15 For we do not have a high priest who is unable to sympathize with our weaknesses, but one who in every respect has been tempted as we are, yet without sin. 16 Let us then with confidence draw near to the throne of grace, that we may receive mercy and find grace to help in time of need. (Heb 4:14-16)
Another way we might deduce that Jesus was indeed without sin, and that apart from even a specific biblical teaching, is if we think it is possible, even likely, that God is an all-perfect being. Or, better said, if God exists, then the God that exists is a maximally great being. And, if there is a maximally great being, then this being must also be maximally good, since goodness is clearly a “great-making attribute.” And finally, if Jesus Christ just is this maximally great, maximally good being, then by force of logic Jesus Christ is also maximally good, and in virtue of being maximally good, Jesus Christ was not capable of sinning, since no maximally good being would be able to sin.
Hope that helps.