ANSWER: This scripture must be taken in context to be clearly understood. First, Paul thanks God in heaven for considering him worthy to serve in the ministry (1Timothy 1:12) "even though in the past I spoke evil of him and persecuted and insulted him." He thanked God again for his mercy and wisdom in knowing that Paul really did not understanding what he was doing (verse 13). He then makes the statement you are referencing.
Christ Jesus came into the world to save sinners. I am the worst of them (1Timothy 1:15)
Paul refers to the time when he was persecuting and killing those who followed Jesus. Yet (he points out) even such a sinner was "shown mercy" in order to show God's patience and love (1Timothy 1:16).
No person lives for any length of time without committing sin. It is a fact of human nature as Paul describes his own plight in his letter to the church in Rome (see Romans 7:18 - 20).
Recognizing that we remain sinners even with our best efforts and striving to live in God’s way brings the humility that comes with understanding of our own nature. God says the person who pleases him is someone who is humble and contrite and pays attention to what he has to say (Isaiah 66:1 - 2).
Because we have free will and independence to choose our own path, no human being is yet "saved" though we hopefully are on the road to eventual salvation. Salvation, in the long term, is escape from the second death (Revelation 20:6,14). There is always the possibility that we will turn our backs on God and choose our own destruction, even though we may presently be on the road to salvation. It certainly is possible to have known the truth of God and been converted to later rejecting God and his ways. Such rejection is referred to in the Bible as the unpardonable sin (see Hebrews 10:26-29, 31).
While we should not be among those who deliberately choose to sin we should be cognizant that though we are on the road to salvation we remain sinners. It shall ever be so until the return of our Lord and Master.