Some people do not believe that moral claims can be facts, or are in any sense true or false. Many more people will claim to be this first kind of person when their beliefs or actions are challenged on moral grounds, but in fact these people believe and act under the presupposition that some kinds of acts are wrong and some are right
People sometimes want to say that my beliefs about the wrongness of using and harming animals are true for me, but do not apply to other people who do not hold these beliefs. This would be odd, because much fewer people believe that my beliefs that racism and sexism are wrong are true only for me. That's why we can disagree with racists and sexists, because we're talking about the same thing. We're talking about whether or not these beliefs (i.e. racist and sexist beliefs) are right, and whether or not actions that are a result of these beliefs is right.
It's also hard to imagine how it could be the case that certain moral beliefs could be true for me and not true for others. I think people say this because they want to find middle ground, and find away that we can be respectful of each other's beliefs. Which is nice, but it means they've misunderstood what is meant in a moral judgment. Moral judgments are necessarily universal; otherwise, they're just preferences. So if you disagree that using animals is wrong, we don't need to shy away from that disagreement. Perhaps in discussing our disagreement, we can come closer to the truth on the matter.
One reason this is difficult for people to understand is that it's not clear what it is we're disagreeing about. Some people think moral disagreement is, in fact, just a mismatch of desires, or conflicting preferences. But this is wrong. On my understanding of morality, the foundational concept is that of reasons, not desires. I'll give an example from Derek Parfit to help explain what I mean.
Imagine a woman in a burning building, standing by a window, with water below her. If she doesn't jump, she will die, and lose what we can be confident will be a very happy and successful life. If she jumps, her life will be saved, and she will be able to live out the rest of her life.
Now, Parfit and I agree that this woman has a reason (indeed, a decisive reason) to jump. The reason is that this action will enable her to survive, and lead a happy and fulfilling life, which is clearly good. She might not want to jump. It may be very frightening, and she might be scared of the water, but she would still have compelling reason so jump. Even if, after careful and thorough deliberation in which she understood all the relevant facts, she decided not to jump, we could still be correct in saying she had reason to jump.
You might still ask what a reason is. The example is intended to show you, to bring out the idea in your mind, which I believe you already have on some level. Parfit, following Scanlon, suggests that a "reason" can be defined as a "consideration that counts in favor of action or belief." Unhelpfully, however, "counts in favor" is best defined as "provides a reason for."
Because, as I am claiming, reasons are foundational in ethics, we should not be surprised that they are hard to define. Many foundational concepts in any field are difficult to define. The concept of a "number" is difficult to define, but this does not prevent us from understanding it, or from making progress in mathematics.
Parfit also asks us to think of the experience of being in terrible agony. The nature of this experience gives us a reason to avoid being in this state. Which is to say, what it is like to be in agony counts in favor of avoiding agony. Is that a true claim? It's hard to see how that could be controversial.
I also think that the nature of animal harm and use counts in favor of refraining from participation in the institution of animal exploitation. That is, I think we have decisive reasons to be vegan. These reasons are much more controversial than those we have for avoiding agony, and much more argument is likely needed to convince others of this fact. But hopefully it is now clearer what I mean when I say that it is true.