First, Celegorm and Curufin held political power in Nargothrond, though Finrod Felagund was the King:
And now Celegorm and Curufin are dwelling in my halls; and though I, Finarfin's son, am King, they have won a strong power in the realm, and lead many of their own people. (The Silmarillion 19)
And one must remember that though Finrod has built Nargothrond, he was but the son of Finwë's youngest son, while Celegorm and Curufin were sons of Finwë's eldest son. And the elves were free to choose to whom they give their allegiance.
But that is not in itself enough to sway Nargothrond. Celegorm then speaks to the people of Nargothrond, and threatens them with the wrath of all Fëanor's sons. Those elves have seen war, they remember the Kinslaying of Alqualondë. Is it any wonder that they are affected?
Then Celegorm arose amid the throng, and drawing his sword he cried: 'Be he friend or foe, whether demon of Morgoth, or Elf, or child of Men, or any other living thing in Arda, neither law, nor love, nor league of hell, nor might of the Valar, nor any power of wizardry, shall defend him from the pursuing hate of Fëanor's sons, if he take or find a Silmaril and keep it. For the Silmarils we alone claim, until the world ends.' (ibid)
And Celegorm was said to be as skilled a speaker as his father:
Many other words he spoke, as potent as were long before in Tirion the words of his father that first inflamed the Noldor to rebellion. (ibid)
If that was not enough, Curufin then threatened Nargothrond with how the quest of the Silmaril would anger Morgoth:
And after Celegorm Curufin spoke, more softly but with no less power, conjuring in the minds of the Elves a vision of war and the ruin of Nargothrond. So great a fear did he set in their hearts that never after until the time of Túrin would any Elf of that realm go into open battle. (ibid)
So there you have it: the elves were afraid to follow Finrod, fearing both the wrath of the Sons of Fëanor, and the war that would come from Morgoth. Celegorm and Curufin offered them the easier choice, and wrapped it in pretty words.
And also, the curse of Mandos was upon them. Finrod says:
if there be any on whom the shadow of our curse has not yet fallen (ibid)
And the curse of Mandos was:
To evil end shall all things turn that they begin well; and by treason of kin unto kin, and the fear of treason, shall this come to pass. The Dispossessed shall they be for ever. (The Silmarillion 9)
You are looking at "treason of kin unto kin", just as was foretold.