You posed two slightly different questions:
A fellow player does not want me to roleplay, what do I do?
- Short answer: Compromise, see other answers for details.
Any way to make him accept roleplaying, or maybe roleplay a bit himself?
I will answer the latter question in more detail. That is, I won't concentrate on how you can compromise to satisfy his curiosity.
First, I'm assuming it is curiosity. It might also be some deeper intersocial problem. He might be trusting neither you nor the DM. If the reason is "untrust", your group has a serious problem to be answered in a different question.
If it's a trust problem, I'd expect the "problem"-player to actively play the rules lawyer. In this case, he should know what the ability you're pointing at does. He would also start at the DM like "That ability doesn't work like that."
I'll further assume the DM and other players are fine with descriptively roleplaying your actions and silently pointing out the intended mechanics. You explicitly said "One of the guys [...] does not like" it. I also assume that your DM thinks pointing is a very clear communication of mechanics and will tell you, if the mechanics do not apply in the way you intend.
Step 1: Deescalate.
I'd politely tell the disagreeing player: "Sorry, but your character has no in game knowledge of my character's underlying abilities. He is welcome to ask, what my character has done or can do. If so, I will answer in place of my character. To not actively encourage metagaming, I will leave it up to you to figure out, what mechanics I'm using. If you think it's important to know in all detail, you're welcome to look, what I'm pointing at."
To stick with your example, your character could answer, if asked:
"I've always been good with animals. Now that I think of, I'm with most people, too. Must be my charming personality, don't you think?"
The thing is, you two don't need to agree on each and every aspect of the game. That is as long as any discussion is short enough, to not disturb the game for everyone else. To get to this point, I'd give the same short explanation once every gaming session, if the issue arises. I'd ignore any further comments on this issue. Not saying anything usually ends discussions the quickest. After all, you don't want to argue. At this point in time you're satisfied, when your gaming style is tolerated. In turn, you should tolerate his gaming style without questioning it or explicitly saying so. Don't say things like "I don't critisize your style!", because that would actually be aggressive.
See how things develop. Probably he stops asking for mechanics, as he gets bored of getting the same answer. Maybe he will start to see the advantage of immersion, but that's unlikely. Take a while (some sessions, depending on the development of situation) before you proceed with:
Step 2: Ask the group.
Well, ask your DM, how he thinks. Ask the group. I'd vote to do it before or after the gaming session. In my group we start with ordering pizza, so I'd discuss it after everyone had the first few bites. Hungry people are dissatisfied people and not to be argued with.
- If your group agrees with his reasoning: Compromise, see other answers,
- If your group is undecided: Try to achieve an explicit mutual acceptance that different gaming styles exist and can be played within your group. If "mutual acceptance" is not enough to strike a deal and constant conflict makes playing unfun, you can try to find a compromise. Maybe printed cards with your abilities in big bold letters (as suggested in a comment of Cryptangel) could be such a compromise. Don't do step 3.
- If a majority of your group (that is not just the loudest or last to speak) shows explicit support for your gaming style, proceed with:
Step 3: Carefully question his approach
So after all, one question remains:
Are you (and the rest of the group) ok with the other player's gaming style? Presumably he makes calls like "I use Charge and Cleave on target B!" If he does so, ask what that means. I personally have no clue what movements are involved to "Cleave" and how to combine it with rushing at the enemy. And which one was target B? Is it the goblin, who I hit over the head? Or is it the goblin's dog that lunged itself at the cleric?
I'd vote for tolerance of both approaches as long as everyone has fun. However, if one approach is questioned by a minority, why not question the other, if a majority opposes it?
Don't push it! He does not like your gaming style. Maybe he will never like it. However he should tolerate it. For a mutual agreement you should tolerate his gaming style, too.
In summary, I proposed the following details to "handle the situation well":
- be polite, invite to in game discussion and offer an alternative to get the exact same info
- keep discussions short to not disturb the rest of the group, although this means not having the last word, which can be quite hard for some people
- mutually tolerate different gaming styles
(Can someone think up a way to lessen any negative impact under the requirement that OP may stick to his gaming style?)