Many of the answers provided to this question have hit on pieces, but so far have not completely connected the dots. The key is the second half of the verse where Paul told Timothy that God did not give them a spirit of fear, but of power.
The gift which Paul gave to Timothy through the laying on of his hands was one of the miraculous powers of the Holy Spirit of prophesy, speaking in foreign languages, healing, etc. all of which miraculous powers enabled the people to believe the words the Apostles and disciples spoke of the gospel of Christ. All of the miracles of the Holy Spirit were given in order to confirm the word.
"...a man approved of God among you by miracles and wonders and signs, which God did by him in the midst of you, ..." (Acts 2:22, KJV)
The power of the Holy Spirit was for the purpose of spreading the gospel of Christ. If Timothy became fearful, and let the scoffers and scorners intimidate him then he would not be able to let the power of the Holy Spirit work.
The prime example that comes to mind is David and Goliath. When confronted with the giant, all of the men of Israel became very afraid and did not trust in God to allow His power and strength to be displayed (1 Sam. 17:11, 24). Until a small boy showed them how God works through the weakness of men.
"David said moreover, The Lord that delivered me out of the paw of the lion, and out of the paw of the bear, he will deliver me out of the hand of this Philistine." (1 Sam. 17: 37, KJV)
David had to show up for the battle, to step forward without fear. Only then could God use him to defeat Goliath, an adversary.
"Fan into flame the gift of God" meant that Timothy should not sit on his hands and let fear rule him, or keep him from going forward. The work of the Spirit to confirm the word could only happen if Timothy went to the battle.
Even though today we do not have the miraculous gifts of the Holy Spirit, we have the word of the Holy Spirit which was confirmed by those miracles in the 1st century AD. The word can only go forward if we boldly speak as the oracles of God (1 Pet. 4:11). (See Note)
And notice the first on the list of condemned in Rev. 21:8,
"But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death." (KJV)
because fear prevents us from doing what is commanded. And, that was the point Paul was making to Timothy. We have to show up for the battle so that God can do His work.
The miraculous gifts of the Holy Spirit poured forth on the day of Pentecost in Acts 2 were time limited.
"For the promise is unto you, and to your children,.." (KJV)
The gifts were not an automatic result of baptism / immersion but from the laying on of the hands of the Apostles or those they appointed such as Timothy. This is evident from Acts 19:1-6 where the Holy Spirit came upon those disciples of Ephesus only after Paul laid his hands on them. Also evident from Peter's interaction with Simon the sorcerer in Acts 8:9-13.
Paul explained that the miraculous gifts of prophesy, speaking foreign languages, etc would fade away in 1 Cor. 13:8ff, but that love / charity would remain. So as the Apostles died off the laying on of hands ceased, and the pouring forth of the miraculous gifts of the Spirit ceased. Today, we have the word in the books of the NT which was confirmed by those miracles.