My 3-year-old cries or throws a tantrum when anyone laughs. She will actually tell us to stop laughing. If it’s strangers she will say to me "mom they’re laughing" while she starts to cry. I would really like some answers on why she does this. I’ve explained to her why people laugh. It didn’t make any difference. The only time I can laugh is if I’m laughing with her.
My oldest son had a similar reaction at that age. From how he developed since (he's now 7), we believe that he was exhibiting some form of social anxiety. He's also very uncomfortable with attention from groups of people, such as having his name announced at a competition or production. He gets easily overwhelmed when presented with a social situation he's unfamiliar with, and we suspect that he either would think people were laughing at him, or that he didn't understand why people were laughing and thus was worried.
The best thing to do in my opinion is to accept her cries as showing a valid emotion, and do your best to help her calm down without denying her feelings. Words like "You don't need to cry" aren't very helpful in this kind of situation; instead, validate her feelings with "You're very upset, what can we do to calm down?" while helping her lower her adrenaline levels and get to the point that you can have a rational discussion of what is going on.
Around 3 she can learn to take deep breaths to manage her emotions; when my son would get overwhelmed this worked (and works!) very well for him. You may need to teach her this skill when she's more in control; resources like this and this for example are readily available online to help you teach her.
As for having a discussion, don't expect a three year old to understand exactly why she is having this reaction; but she can probably tell you some of the what, and perhaps that will help you explain to her the why. Does the laughing scare her? (Maybe it's a loud sound that she is uncomfortable with.) Does she think they are laughing at her? Does she understand why they are laughing about? Ask probing questions that she can answer, and then focus on giving her tools to manage the stress - again, don't tell her not to have the stress, the emotions or stress simply are; focus on how to manage those feelings or emotions when they overwhelm her.
When my son felt this way, we weren't necessarily able to find out all of the issues he was having, but he did tend to say that he didn't like the sound (he was bothered by loud noises) and that he didn't think anything was funny (which was consistent with his very self-centric worldview, that if it wasn't funny to him then it wasn't funny). We guessed that it was also his inability to understand the social dynamic, and so tried to help him understand why people were laughing, always after the emotion had passed. If I remember properly this lasted around six months or so until he either learned to cope with it or developed enough emotionally that he was no longer worried about it.