Aside from using it when sizing up their opponents, Saiya-jins use power level to determine the relative strength of their own children at birth so as to classify them within their caste-based social order. Since they are a warrior race, this is how they determine who is valuable and who is not.
Vegeta and Nappa were both born into royal families. Vegeta informs Goku that he was born into a low caste and sent off to another planet because of his low power level. Vegeta is the royal prince and strongest of his race, and Nappa is the general of the entire Saiya-jin army. The fact that Goku, a low-caste Saiya-jin, now has a power level over 8,000 is a slap in the face to Vegeta and Nappa.
Nappa has a power level of 4,000 and Vegeta has a power level of 18,000. This is why Nappa freaks out while Goku remains confident. It’s why Vegeta is so upset and screams “It’s over 8,000!” and then smashes his Scouter. Vegeta isn’t afraid, he’s just especially surprised and annoyed by Goku’s rapid growth, because less than a year earlier, Goku had a power level of only 334 while fighting against Raditz. And how could a lowclass Saiya-jin have such power? Goku’s incredible change was a result of his training in the afterlife with North Kaiō, master of the northern quadrant of the universe.
In Dragon Ball Kai, Nappa gets beaten down and punched into a small mountain by Goku. He yells, “Damn you to Hell!” to which Goku calmly responds, “I see now. As expected, you’re not that tough.” Nappa shouts back,
“I’m an elite warrior from a noble family! I won’t allow some low-class trash like you to make fun of nobility like me! You’re gonna’ pay! I’ll freakin’ kill you!” Nappa’s self image and pride are threatened by Goku’s existence, and in order to validate himself, Nappa has to kill this “low-class trash.” In Dragon Ball Z, a warrior’s pride becomes a matter of life and death.