Imagine getting married, and during the wedding a pivotal moment in the ceremony came when your spouse-to-be placed an ankle bracelet on you with a GPS tracker so as to make sure you would remain faithful to them in the coming weeks and months. Because they could not bring themselves to extend any trust to you, they wanted proof, right from the beginning, that you'd follow through on your vows. Such a scenario seems outrageous to us because we cannot imagine entering into a relationship with someone who will not extend any trust on the front end that can be built over time.
...Yet this is precisely what we expect God to do with us. Although we are made in his "image," and therefore derive aspects of relatability from him, we still expect him to enter into a relationship with us though we'll extend no trust to him. We would find this totally unreasonable, and yet we expect that he's cool with it because he's God and should be above such petty relational dynamics like trust, confidence and faith. On the contrary, our intuitive requirement of trust to enter a relationship is a reflection of his requirement. Without trust in his saving work and benevolent care we cannot expect God's relatability.
For this reason trust and proof can run contrary to one another. Proof can be offered from the one who is trusted (out of love for the one who is trusting), but proof that is demanded evidences an absence of trust. I can tell my wife where I've been most of the night (if pastoral or chaplaincy duties called me away) as an expression of love, rewarding her trust in me. However, if she demands to know where I've been, she's evidencing a mistrust of my behavior during the time I was away. Out of love for her, I'll also avoid behaviors that erode her trust. This is a subtle dance of proof and trust, but it makes sense in relational terms.
Another term for trust, but conveys the same meaning is faith. When a relationship begins, faith in the other person's trustworthiness must be assumed, otherwise the relationship has not begun well. Faith grows in the relationship as those in the relationship prove faithful through new and diverse experiences. Though faith grows with time, it must have been present in the beginning. It is the same with God. Our first response to his invitation to relate to him through Jesus Christ must be faith. While some evidences are graciously offered by God that can point toward the reasonability of faith, reason and evidence can never substitute for faith. What one trusts God for is salvation from spiritual death and judgment in Hell; realities that can never be proved, only believed in.
Faith is scary though. It is the acceptance of truths about reality that cannot be proved regardless of how much "proof" is ever surfaced. Faith cannot be substituted by good deeds, as though God needs to be convinced of our faithfulness. The contrary is true - we must be convinced of his faithfulness. Faith cannot be circumvented by good deeds, good upbringing or good company. Faith cannot be made unnecessary by being a church member, a good neighbor or a Republican. Faith cannot be usurped by giving to the church, receiving the sacraments or even singing with enthusiasm. Faith alone is the basis of our response to the work of Jesus Christ.
Faith alone acknowledges the relational requirements of God on us; that our only correct response is to trust him for all that we cannot see, especially since he has shown himself so faithful with all that we can see. Nevertheless, faith alone is each person's proper response to God's revelation. This cannot be replaced with sacraments, good deeds, good behaviors or positive thinking. Faith alone assures us that the relationship with God through Christ is secure. Sola fide ("only faith") is a necessary cry in a world where people still think that God is impressed with "good" people. On the contrary, he's drawn to people who trust him, approaching him only with their faith in his promises...nothing more.