Hebrews 11:6 And without faith it is impossible to please him, for whoever would draw near to God must believe that he exists and that he rewards those who seek him.
Believing in God is not just about believing that He exists, but also about believing that there is therefore a way that we ought to live. In other words, the way to believe in God is by believing that we ought to be a doer of His character traits, so the significance of our good works is not that it is part of something that we are required to have done first in order to earn our salvation as the result, but rather it is the way to have faith in who God is to be the guide for how we ought to live our lives, and it is by that faith alone that we are being saved. For example by being a doer of good works in obedience to God's law we are testifying about God's goodness, which is why our good works bring glory to Him (Matthew 4:15-23), and by testifying about God's goodness we are also expressing the belief that God is good, or in other words, we are believing in Him, and the same is true for God's other character traits.
This is also the way to believe in the Son, who is the radiance of God's glory and the exact image of God's character (Hebrews 1:3), which he expressed by setting a sinless example for us to follow of how to walk in obedience to God's law. In other words, the Son is the living embodiment of holiness, righteousness, goodness, justice, mercy, faithfulness, love, joy, peace, patience, kindness, gentleness, self-control, and so forth for other aspects of God's character, and the way to believe in the Son is by being a doer of his character traits in accordance with his example. In Hebrews 11, it lists examples of people who had saving faith in spite of having limit knowledge of the incarnation of the Son who nevertheless had access to the Father through the Son by embodying his character traits.
This is also why there are many verses that connect our faith/belief/trust in God with our obedience to Him. In Revelation 14:12, those who kept faith in Jesus are the same as those who kept God's commandments. In Proverbs 3:5-7, we have a choice between leaning on our own understanding of right and wrong by doing what is right in our own eyes or trusting in God with all of our heart to correctly divide between right and wrong by being an doer of His instructions in all of our ways and He will make our way straight.
The content of a gift can be the experience of doing something, such as giving someone the opportunity to experience driving a Ferrari for an hour, where the gift requires them to do the work of driving it in order to have that experience, but where doing that work contributes nothing to detract from the fact that the opportunity to drive it was completely given as a gift. Similarly, the content of God's gift of eternal life is the experience of knowing Him and Jesus (John 17:3), which is through being a doer of His character traits, and the gift of God's law is His instructions for how to have that experience (Exodus 33:13, Matthew 723), which is also why Jesus said that the way to inherit eternal life is by obeying it (Luke 10:25-28, Matthew 19:17).
Sin is what is contrary to God's character, such as with righteousness being in accordance with God's character while unrighteousness is sin and sin is the transgression of God's law because it was giving to divide between what is in accordance with our contrary to God's character. Our salvation from sin would be incomplete if we were only saved from the penalty of our sin while we still continued to live in sin, so there must be an aspect of the gift of our salvation that we are experiencing in the present by being a doer of God's law, which again is what Titus 2:11-14 is describing.