To quench is to put out or extinguish and we are asked to take up the shield of faith (above all) wherewith we shall be able to quench (put out) the fiery darts of the devil. It is however, important to recognize that we do not put out the fiery darts (trials and temptations) but instead we need to merely take up the shield of faith, that protects us against these darts. It is also important to recognize that we ought not to quench (put out) the Holy Spirit of God that dwells in us (1 Thessalonians 5:19). For when we quench the Holy Spirit, the flesh that is at war with the Spirit triumphs and makes us sin that leads to death, trampling the Son of God and nullifying his sacrifice (Hebrews 10:26-29). So as Christ Ones we ought to quench the fiery darts and also quench not the Holy Spirit.

Ephesians 6:16 (KJV)
16 Above all, taking the shield of faith, wherewith ye shall be able to quench all the fiery darts of the wicked.

1 Thessalonians 5:19 (KJV)
19 Quench not the Spirit.