You need to do some reading up on game design, progression systems and why players play games... the whole point of making a game is usually to give the player something do, something to achieve. Why is your game worth playing? etc etc... Outside the challenge of your gameplay, "achievements" are a direct way to give your player something to strive for. Usually these are goals outside the normal gameplay. Although some achievements are as simple as: "Complete the game" (because maybe you have a particularly long game and want to give incentive to finish it). Other than that you can sometimes get players to replay your game if you give them certain challenges. i.e. "Complete Level 1 with only using 20 bullets" or "Complete ALL other achievements in the game" etc... the possibilities are endless.
Giving achievements can provide an actual physical dopamine rush for your player... which is sometimes incentive enough for some players. Others it's a point of pride, and others it's a matter of having to finish every nook and cranny of a game. I think most players won't care much at all about achievements (depending on the genre of your game).
So do they make your game better? I don't know about that... will they make you more revenue? not directly. But if your game is good, achievements could make players play it again, could make them like the game further and talk about it to others (remember those prideful players) which could indirectly turn into word-of-mouth --- giving you more sales.