Vegeta has come a long way since his introduction in Dragon Ball Z. He went from being the prince of a fallen race and an enemy of the Z Fighters to a beloved member of the group after helping Gohan, Goku and Krillin defeat the galactic tyrant Frieza on Namek -- even giving up his life to do so.
While Vegeta had always planned on taking down Frieza in revenge for killing off the Saiyan race and his father, he actually assisted Goku, whom he had tried to kill only roughly two months before. Since being wished back to life that first time, he's learned many things regarding his own abilities and strength. Arguably one of the most important was the ability to sense power levels without a scouter.
Being a member of the Frieza Force, Vegeta was equipped with the signature Scouter, an eye-piece that could read and estimate a being's spiritual strength and power. This was mostly a plot device Akira Toriyama used to help readers understand who was stronger in a more visual way. Because the Scouter would scan the amount of ki someone let off, it was easy to fool the user into believing an opponent was weaker than they truly were by hiding their energy.
The idea of power levels was first implemented was in the original run of Dragon Ball, as Goku learned how to sense ki after drinking the Ultra Divine Water (aka Mystic Water or Super Kami Water). It was hinted that Master Roshi could do it earlier than this, when he couldn't sense General Blue before the group was ambushed. After he was defeated by King Piccolo, Goku went to Korin and asked for a way to help avenge the death of his friends. The other fighters would eventually learn to sense ki too, but it's never clear if they also drank the water or if Goku taught it to them.
As a member of the Frieza Force, Vegeta wore a Scouter like Raditz to learn his opponents' power levels. Once he met with the Z fighters on Earth, he quickly realized the little devices were pointless as his opponents could manipulate their energy output, making the Scouter "most unreliable." Habitual use would only be a hindrance in their fight, so Vegeta and Nappa removed them so they could no longer read the numbers. In other words, they learned from Raditz's mistake.
Vegeta would also go on to learn how to sense power levels without a Scouter, making it difficult for Frieza and his flunkies to follow him on planet Namek. How he learned to do it in such a short time is unknown, but it was one of his biggest breakthroughs as a fighter. The Saiyan Prince didn't need to rely on a computer that more or less acted as a representation of Frieza's power over him.
By learning to rely on his own skills -- rather than technology that could do it on an often-less accurate degree -- Vegeta was one step closer to getting out from under Frieza's control. Sensing energy makes it easier to determine if something more powerful than an average human is coming. Conversely, the user can also track down a powerful being anywhere in the universe by using their ki energy.
Being able to feel how strong someone is allows Vegeta to determine whether it's worth using his energy to fight, or to retreat and come up with a different plan. If his other senses aren't able to see or hear something, then he can always use power-sensing to find an opponent. Strength is important in Dragon Ball, but knowing how to combine it with other skills will prove to be even better.