Paul is reminding Timothy that he is not there of his own will, but of God's will; to spread the good news of Christ.
To fight the good fight means ,to me, to stand firm im the truth of Christ's love and God's forgiveness. Not to be shaken by the world around me or the forces that may try to knock me down.
Paul is encouragind Timothy to stay strong. Even though things seem hopeless, hope is never lost.
Paul is trying to counter the temptation to go along with the crowd. It would be easy for Timothy, in his youth, to go along with what is popular.
This is a reminder to us all to stand firm in our faith, regardless of what is considered popular. The enemy tries to trick us every day into doing what is not right, we need to remember that even when things are hard, God is with us. As long as we stand firm, we will be safe in his arms.