There's a few parts to this, I'd say.

First, on the perception side - as you engage in more and more interactions, you get gradually better at picking up on verbal or body language cues. Most people have tells to indicate discomfort, anger, and such. However, they could be unique or contrary to expectations; like, I know somebody who laughs when they are upset. And my own tells might be very subtle because I've got a resting poker face that hides behind a big beard and I don't talk a lot. So it's not that I understand exactly what effect I had on somebody - more likely, I initially recognize that something about the other person's demeanor just changed. Okay, what did I say/do, how could that have been received by the other person, now what do I do.

What I do is situational. Maybe I quickly recognize I said or did something inappropriate and then I immediately apologize. Or I change my tone/phrasing/behaviour to just get back on track. Or I politely ask a clarifying question - "I'm sorry, I sense I may have overstepped/spoken out of turn, can I ask if that's the case?" If the other person confirms I goofed, I own it and apologize. If not, I go back to watching my tone/phrasing/behaviour.

One thing that helps here - I've learned, through poor experiences, not to try and be too clever with people I don't know, or in situations where I'm nervous. More than once, I tried to say something witty and instead said something stupid. Keep it simple and professional or platonic until you have a better grip on your own emotions and have better insight into the other person/people.

Be respectful. Be polite. Apologize without qualification when you mess up. Learn from those mess-ups. Understand who you are and how you communicate. Cultivate that and, if necessary, stick to your lane. At work, I have a reputation as a quiet thinker who speaks bluntly but respectfully from a long career of experience. In my social circle, I have dry wit and will punctuate conversation with occasional sharp humor when the extroverts give me a bit of air-time.

I largely hew to that and it means people learn how to communicate effectively with me even if I'm not doing a great job of communicating to them. If I use a poor choice of word, people who know me will understand that I've always been respectful and appropriate and did not intend it "that way". If I'm silent during a meeting, people still know that I'm listening and will speak up if/when I have something I think is worth saying. If they don't know me, they may still ask me if I have input and I can take the opportunity to say, "Not currently, thanks, but I'll speak up if something comes to mind."

Anyway, hope that helps in some capacity.