The first step is to look for the most obvious signs. (Sounds trivial, but many people don't it, instead they create overcomplicated hypotheses.) If someone smiles, they are probably happy. If someone screams, they are probably angry. If someone cries, they are probably sad. This is not going to be 100% perfect, e.g. sometimes people smile when they feel embarassed. But it's a good starting point that you shouldn't skip.
(Many people already have a story about how the other person feels, and either ignore the signs or try to find an explanation within the story. For example, I assume that the person is angry... but the person smiles at me... what a hypocrite!! Instead of considering that maybe I was wrong about the person being angry.)
For someone on the spectrum, probably the most difficult part is to stop doing what you are doing now, and observe the other person curiously. And make the habit to do this as often as possible. Because this is often more important than finishing the sentence (paragraphs, the entire speech) that you already started. This will be really difficult -- but the encouraging thing is that when you master this skill, you can become better at it than many neurotypicals. Because they are doing it in an instinctive half-assed way, but when you learn to focus, you will focus intensely. (Of course, a neurotypical could learn the same thing, the same way, they just have less motivation to do so, because they assume they are already perfect at it, so most won't bother.) Knowing that you don't know is a good start.
And by the way, neurotypicals also fail at empathy quite often, they just cover it better, and recover from a blunder faster. They kinda do it in a way that is more socially acceptable. For example, they sometimes misinterpret another person in a way that makes the other person go "wtf?", but all onlookers go "wow, that was an awesome example of empathy". While you, if you misinterpret someone, that person goes "wtf?" and everyone else goes "wtf?" too. Maybe the proper way to put it is that neurotypicals are very good at reading the feelings of a crowd, but it is hit-and-miss with reading the feelings of one specific person. On the spectrum, it is hit-and-miss with one person, and almost sure miss with a crowd.
Also, look at the compatibility of their words and actions. If someone says "I am okay", but they keep yelling and hitting things, they are probably not okay. But confronting them about it directly is not helpful. Maybe later, if they calm down, you could ask them "yesterday, when you said you were okay, you also..., so I am confused". Here it depends. Some people will explain. Some will get angry again; in that case, don't try this with the same person again, find someone else to talk to.
For long-term understanding of other people, focus on what they do, instead of what they say. The past behavior is the best predictor of future behavior. If someone keeps talking about how they are going to do this or that, but they never do it, just assume they never will. Someone keeps doing something, then says excuses and promises to never do it again, but keeps doing it, just assume that they will keep doing it. The words can provide additional information to actions (e.g. "I hit you because I was angry"), but cannot erase the action (whatever was the reason, it is a fact that the person hit you, and you should not forget that, especially if that happens repeatedly). Words are "maybe", repeated actions are "the real thing". (Repeated words do not become any more reliable. Unless you mean "reliable prediction of what the person will say the next time", of course.)
People are different. Some are liars, some are manipulators, some are confused, some are sincere. Sometimes your problems with understanding a specific person are a result of that specific person being difficult to understand. Don't worry too much about understanding one specific person; pay attention to understanding people in general. It is possible that your understanding of other people will gradually improve, and yet that one person will remain just as mysterious as before. Count that as a success, not as failure.
People are more similar to each other than it may seem. https://en.wikipedia.org/wiki/Barnum_effect Most people are insecure, with various degree of denial. (Major exception: psychopaths. Charming people who usually refuse to discuss what is inside them, because there is barely anything beyond emptiness and hunger.)
Like all skills, empathy improves with practice. It would probably be useful if you could find someone to observe you at social situations, and discuss with them the situations afterwards. Probably more than one person, because each person will have their own biases and projections.
Maybe try to read some books or watch videos on Rogerian therapy. Simply said, it is a way to communicate with people in a therapeutical setting without imposing your own ideas of how they feel. (If you don't know how they feel, and you know that you don't know, that should make it even easier.)
This is just a random stream of thought, I hope it is useful, and I definitely have no scientific evidence for any of that. Good luck!