There are already some good answers. In particular I agree with the advice to arrange for a break, but I want to spend some time on why telling someone on the opposite side of an argument to "calm down" is usually disastrous. This can be useful for finding better wording, or at least avoiding poor wording. Even calling for a break can run into similar issues if you are not careful, e.g: "Let's all take five minutes to cool off." That phrase can easily come off as a passive-aggressive way of saying "calm down".
So, what's wrong with saying "calm down"?
First, it either is or is easily perceivable as being presumptuous. If the person is quite visibly angry, for example they are yelling and/or have aggressive body language (hitting things, aggressively leaning in), then your external assessment of their emotional state is quite possibly more accurate than their own self-assessment in that moment but only because they aren't in a rational frame of mind. If they don't agree with your assessment, then it will be perceived as presumptuous. If you only "get the impression" that they are angry, then you are literally presuming their emotional state. In the context that the comment you quoted was given, this was dramatically the case as it was an email exchange.
Following and compounding that, "calm down" is an order. It's grammatically an imperative statement, and semantically it's a command. This is why it is "especially [rude and condescending] if they are your senior" and why Captain Emacs in a comment here states that it "[may] well be a dominance display". This is bad enough on its own, but (even as a subordinate) if I don't agree with the presumption that I'm angry, how can I comply with "calm down"? Heck, removing all the social dynamics aspects, "calm down" just isn't a very actionable command. It's like telling someone to "work harder". What does that mean? What do you actually want them to do? How will they know they are "working hard"? What if I don't know what to do to "work harder"/"calm down"? This is one of the benefits of "let's take a break"; it clearly specifies what to do and is a good strategy (even for yourself) for calming down.
Further compounding, why should they calm down? There are two cases: either they agree with you that they are angry, or they don't. In the former case, "calm down" comes off as "you being angry inconveniences/displeases me, so stop". This is a rather uncharitable reading, but there is little reason to expect things to be taken charitably in such a situation. In the latter case, they have to wonder what it is that they are doing that you see as indicating that they are angry. There's an apparent possibility. Apparently, the very act of disagreeing or arguing against you is perceived by you to be indicative of anger. Therefore, by saying "calm down" you are really saying "stop disagreeing with/arguing against me". Further, you may be trying to frame them as "the unreasonable one". Again, this is a somewhat uncharitable reading but is a likely consequence of the presumably oppositional situation and the vagueness of what constitutes "calming down". And frankly it may well not be in their interest to calm down. Applying this to calling for a break suggests against wording like: "Let's take a break so we can have a more productive discussion." "Productive discussion" comes off as code for "a discussion in which you agree with me".
Here are some potential tips on what to do, as opposed to what not to do. In a meeting setting, you may have some procedural rules you can apply, e.g. if discussion on a topic takes more than five minutes it automatically gets tabled. Even without such a rule, saying something like: "It doesn't seem like we're going to reach agreement on this soon, let's table this until after the other agenda items/let's schedule a separate meeting for this/let's talk about this one-on-one later" is an option. It is important to to have that follow-up aspect. For a more one-on-one situation, you can use your own emotional state to call for a break. For example something like: "Can you give me a few minutes to clear my head/collect my thoughts?" or even "I'm starting to get worked up, can I take five minutes to calm down so I can think about this rationally?" (I don't advocate the latter, unless it's an accurate statement of your internal state.)
Of course, to not just get back into a heated argument you need to first calm down yourself if necessary then reflect on what they were saying, what you might have said/done that prompted the reaction, whether your reading of the situation is accurate, what to say or do instead, if this "fight" is worth it, and even whether them "staying calm" is actually important to you or in their interest. For example, I'm reading Thomas Schelling's "The Strategy of Conflict" right now and he discusses how demonstrably curtailing your own rationality can be a rational strategy. Obviously figuring this out can be difficult which is why I find books like Schelling's, or Edgar Schein's "Helping" to be interesting and valuable.