Your question is a bit vague, so I'll answer the question I think you're asking ("How do I explain to meat-eaters that I'd rather not cook/handle/buy meat/animal products for ethical reasons without being rude?").
First I think we should define "rude" as some people will take offense at the mere mention of your choice to abstain from the meat industry in all ways possible. The line between explaining a choice and being rude is when the explainer shames the other party for not making the same choice as they have.
In order to avoid being objectively rude, the explanation of your refusal to handle meat should stay centered on you and your choices not on the person who made the request and theirs (i.e. "Handling meat makes me uncomfortable, so I'd rather not cook that steak for you." vs. "Eating meat is disgusting and immoral, how can you support such a terrible industry?!"). If the recipient of this explanation is not satisfied with you saying it makes you uncomfortable and is dead set on pushing your boundaries, only then should you go deeper into your reasoning for abstaining from handling animal products. Even then, you should keep the focus on yourself (i.e. "Some people volunteer abroad, or build houses for the homeless, abstaining from animal products and the industry as a whole is my way of doing good in the world and I have a personal boundary when it comes to buying/handling animal products as I would be supporting that industry.").