Disclaimer: I'm not a professional psychologist. I try my best to give an undistorted view that is backed-up by psychological studies. Please take this post with a grain of salt.
I read about psychological topics in the quest of improving my knowledge of human nature. I do this to understand social situations, emotions and people better. This improves my ability to interact with a big variety of people and enables a deeper understanding of their problems and what they want to solve. It also leads to better communication and negotiation skills and you can improve your understanding of your customers.
So it's easy to see why this skill is very useful. To train it, we have to focus on ourselves first. "Why care about ourselves when we want to understand others?", you may ask. The reason for that is that our perception shapes our reality and having an inaccurate model due to cognitive biases decreases our accuracy to analyze social situations. We can't completely eliminate this bias, but we can reduce it through self-reflection, discipline and a habit of scepsis with regards to our own perception.
I will now explain how to make a more accurate analysis in social situations utilizing this knowledge.
start with the emotion
You need emotion, but you also need to understand the emotion. Problem is that your limbic system can't talk (this area of your brain is responsible to create emotions). Therefore, we resort to rationalizations - explanations for our behavior that we basically get through guessing. But most of the time, the rationalization is highly inaccurate. Why? Because our unconscious side has a lot of information about negative things about us and we try to avoid it because it doesn't help us feel good (this emotion exists solely so that mental health can be preserved and cognitive dissonance gets reduced).
You are angry that your husband is late.
Possible explanation for your anger:
You are angry because he's always acting selfish and doesn't respect you.
Other explanation that is probably more accurate but contains negative aspects that the consciousness hides (drive attention away from) to reduce cognitive dissonance:
There is a deep fear to get rejected. Maybe there are experiences in the past where trusted people just left you. This deeply rooted fear triggers anger as a reaction because your unconscious side sees a threat (indirect increase of likelihood of death through social rejection).
How to utilize this knowledge?
Analyze your emotions and try to ask: "What useful information does it contain?" and "What are the root causes for this emotion, what is the incentive system behind it?". The hardest part is that you have to know about your deepest insecurities, deficits, negative experiences and fears ("I'm objectively not very attractive", "My mother never showed me her love in unconditional ways", ...) and regulate your emotions (instead of acting out that anger, you analyze it on a meta-level and extract information from it for a more accurate analysis). This requires extreme self-reflection and years of struggling. When one finally masters this, s/he's considered enlightened which basically means that s/he's able to not only analyze her/his own behavior and emotions, but can also use this knowledge to act accordingly.
analyzing social situations
Maybe you've witnessed the situation in the past or heard about it: A guy and a girl have a date. First, they talk about superficial things and begin with small talk and slowly progress to deeper topics and start to talk about work, joy, happiness and their life goals. From an outside perspective, you can clearly see that he is more engaged in the conversation than she is. Her body language is expressing clear disinterest while he enthusiastically talks about his last holidays, but for some reason he doesn't read the cues she's sending. After the date, he tells us that "this was a great date, there's clearly something going on between us", while she is completely indifferent. Why is that?
Because his cognitive resources are mostly used to vividly remember his holidays, he can't concentrate on her while she's able to analyze him unconsciously. All aspects like jaw line, odor, his height, the sound of his laugh, the way he talks, the way he moves his arms and his body posture. All of this gets analyzed simultaneously. This happens mostly unconscious and results in an emotional reaction e.g. that she feels uncomfortable around him. She will send out subtle cues to indicate this e.g. reaching to her throat and changing the direction of her body.
But the thing is: He needs to have cognitive resources to understand and analyze those cues and decode them to understand her emotional state. It's not because he's unwilling to do this, but because it needs mental energy he currently uses for different tasks. If you want to learn more about this, I can recommend Daniel Kahneman's "Thinking, Fast and Slow" (although many of the cited studies are not replicable).
Another problem is that he would have to admit that she clearly isn't interested. If you have a low self-esteem, the fear of rejection can induce a feeling of anxiety that you want to avoid. It's therefore very possible that his subconsciousness hides those clear cues (which show that she doesn't like him) from his consciousness. This cognitive bias exists to ensure that his self-image doesn't get damaged and to reduce cognitive dissonance. If you want to learn about the ways our own consciousness tricks ourselves, read about cognitive biases - the Wikipedia article has a pretty big list.
It's also important to understand that cognitive biases are not necessarily bad. They are good heuristics that help us to analyze a big amount of data (i.e. social situations) in a fraction of seconds. But they are inherently inaccurate and we can correct their estimates using knowledge about ourselves and cognitive biases to improve their accuracy. There are some cognitive biases e.g. hindsight bias that are hard to correct even if you know about them, so this is not a cure-all. Another thing is that you need enough cognitive resources (which sounds very abstract but can be measured using brain scans i.e. brain metabolism).
what you'll get
This is a truly painful and never-ending process, I have to admit. But the goal is to improve compassion and your Theory of Mind. If you understand your unconscious side better, you can understand others better because your mind won't get in the way of seeing the things clearly - it won't be possible to achieve this state in a pure form, but you can improve this ability like everything else using knowledge, training and habit-building.
Another positive side effect of this is that you develop a thick skin and you can't get hurt easily through words. If you know your weaknesses and face them, no one can harm you with them. Most people have deep insecurities that they aren't open about. This creates a huge area where people can hurt you. It's not about hiding those weaknesses, it's about accepting yourself. So another positive effect is that you can develop a strong self-confidence through this process. All those aspects help you to be a stronger person. You don't need to show fake confidence using humiliation or insults and don't become anxious when somebody doesn't like you or reacts negatively towards you.
This post is extremely simplified, but I think it gives a better introduction into the world of cognitive biases than referencing a bunch of psychological studies. If you're interested in more, I can recommend "Psychology" by P.G. Zimbardo (must-read for psychology students), "Thinking, Fast and Slow" by D. Kahneman, "The Invisible Gorilla" by Simons and Chabris (especially funny - I've tried this with a school class and were amazed that half of them didn't notice the gorilla), all types of book about biopsychology, evolutionary psychology, computational psychology (especially for computer scientists), communication books (e.g. about communication models and the different aspects of communication) and a lot of other resources. I've also enjoyed the Unified Theory of Psychology where a psychologist tries to embed the current knowledge in psychology in a coherent framework.