One of the places we get stuck is when we don't fully understand the experience of the injured partner. When working with couples I tell them that the ace up their sleeve is empathy. If you can show your partner that you understand how they feel, without making the mistake of trying to simultaneously explain or, worse, defend your position, they will melt like butter.
There is brain science behind this. When both of you are in a heightened state it is much easier to see the other as a threat. So when you try to present your side of the story, if you have not first calmed your partner down, they are NOT going to be listening! If you can connect to your partner using the social engagement system of the brain, including mirror neurons, and create a feeling of syncrony between your two brains, then you have a much better chance of being heard. CALM YOUR PARTNER'S BRAIN before you try to tell your side of things. For instance, in a heated argument even if you manage to convince your partner that you are right through your brilliant arguing skills, he or she is likely to later feel bullied or disregarded and hold that grudge. So you really haven't won anything. Much better to stop trying to prove your point, empathize, and then make a repair. Once your partner's brain is back together (see "Flippin' Your Lid") and they have invited you to share your side you can express your opinion. And since you did such a great job in helping your partner to feel heard they are likely to reciprocate the favor.
So how exactly does one achieve this mysterious thing called "repair"? Well, as mentioned already the first thing is to STOP TRYING TO PROVE YOUR POINT. Set that agenda aside. Then look at your partner (this is really done best in person and within the attachment distance -- 18 inches or less) and tell them, as best you can, what you understand are their feelings about what has happened. For example, if the fight was about you coming home from work late, which s/he has told you drives them crazy, you could say "I know it really seems to bother you when I come home late and I know that last night I got home an hour after I said I would. I could tell that you were really upset. I wonder if ___________ (and insert your best guess as to why it bothers them, such as "you were a latch key kid and so waiting at home alone feels really lonely and sad). Or if you don't have a clue as to why they have such a strong reaction, say "I really want to understand why this bothers you. Can you help me understand that?". Say this with your most loving voice (human's threat appraisal system is very attuned to pitch, tone and volume). Also make as much eye contact as you can muster (this makes the message get in on a deeper level of the brain).
If you are too grumpy to jump right in to this part, then tell your beloved "hey, I want to work this out but I am really activated/triggered/upset right now and I need to go away for _____ minutes (no more than a few hours). I will come back at ____ and we can talk it through. Then your job is to go somewhere and GET REGULATED. This is not just time for you to take a break and go watch TV or hang out with friends. You have a job to do, which is to get YOUR brain back on track. So go meditate, exercise, be outside in nature, listen to calming music, practice a relaxation technique or whatever helps you let go of stress.
Now once you are calm go initiate the repair and empathize. Tell your partner what you have observed and show your desire to really feel what it's like to be in their head/heart. Once you have done that it's time to apologize. I know, I know, maybe you didn't really do anything that you thought was wrong. But you can still apologize for hurting your partner's feelings. Say something like "I am so sorry that I caused you pain. You are the most important person in my world and I really don't want to hurt you." Remember, if you do this right your turn to tell your side of things will come very soon! Hang in there.
It's VERY important to "check your work" after you have attempted to make a repair. Look at your partner's face and body. Do you see them relax? If you have done your job you should see a softening of the face, relaxing of the neck and shoulders, maybe even a big sigh. If your partner still looks like s/he wants your head on a spike then you need to double back and keep trying. Once you get good at this you will be amazed at how quickly you can de-escalate your partner and get back to "friendly" status.
The pay off of learning to do a good repair is multifaceted. Your partner will love you more for working so hard to understand him/her. You will feel proud and accomplished that you have learned how to "tame your lion". You will be able to enjoy your partner feeling loving towards you again instead of worrying they are going to kill you in your sleep. And you can salvage the rest of the evening/weekend/vacation without having to have the fight ruin things. Most importantly though you stop this event from going in to long-term storage and becoming another brick in the wall between the two of you or ammunition to be brought up months or years later when they are wanting to exit the relationship.
Remember, most relationships don't implode over night. It's the slow accumulation of misunderstandings, unrepaired fights, feelings of being dismissed or disrespected that over time kill most of our failed relationships. Taking care of business right away and doing a good job of it is practicing good relationship upkeep.
Best of luck to you and yours in your relationship journey.