Finishing under pressure. It is what separates good players from great players. Unlike other sports, tennis players have to win the last point, they have to finish. If a basketball team is up by 20 points with a minute to play, they do not need to keep hustling on defense or keep executing on offense (although the coaching staff would certainly want them to do both). The player in the lead is playing with all the pressure, while the player who is behind is playing pressure-free. Have you ever noticed how your opponent plays the best point of the match when behind set/match point? As Agassi pointed out in his book, there is an unstoppable force that exists when nearing the end of the match; either a force that is pulling you closer to the finish line, or the force that keeps you from it. When the force pulls you closer everything is going well, you can do no wrong. But when the force is against you it feels like nothing you do will get you closer, you can do nothing right.
So how do the best players in the world finish? In the 1988 US Open finals, Mats Wilander said he served and volleyed on match point because his hands were shaking so much from the nerves; he knew he would tighten up even more if he played from the baseline. Todd Martin worked on getting his thoughts more structured and present-focused in the months after his loss to Mal Washington at Wimbledon (Martin held a 5-1 lead in the 5th set before losing 10-8; he had two chances to serve it out). Martin began seeing himself play against Richard Krajicek in the finals, and ultimately he lost his focus and choked. To answer the question above...the pros work on it. They get nervous just like you do, they get tight in big moments just like you do. But they get to work on developing the skills to overcome the nerves and this is what separates them from their peers. They find effective ways to manage the nerves and the pressure, and quite frankly, they learn to handle it better than everyone else. As famed tennis coach Chuck Kriese always says, "Pressure is a privilege." To be in a position where you feel pressure means you have done something well to get to that point. You are in a position to win. Finding your own answer requires 1) awareness of which situations bring about the most pressure, 2) trying different strategies to better deal with this pressure, and 3) being in a position to win and face the pressure.
Approach the end of matches with a new mindset and arm yourself with new ways to perform under pressure. Your opponent will usually play better when behind, so it will take you raising/maintaining your level to finish. Here are a few ideas to implement in your own game: 1) keep your intensity higher than your opponent's with positive reactions and a strong walk, 2) focus on how you like to play and win with your strengths, 3) do the work in between points by executing your routines and relaxation skills, 4) reframe how you look at pressure and change the dialogue in your head (from "I am so tight", to "I've earned this pressure") and 5) fully commit to each shot (hesitation or second-guessing only increases the nerves). Some players never learn what pressure is all about because they are rarely in a position to win. Struggling to finish matches is actually a great sign, as it means you have put yourself in a position to win. But to go from good to great you will have to eventually find what works for you when faced with these situations. Earn the pressure and persevere.