You analyzed the situation quite well already. Now think on how to resolve this well for the long term!
As you said, the tantrums won't go away by giving in, just because the neighbours said so. Your plan to talk to the little guy, once calm again, is a good one. To speed up the cool down, stay calm yourself, and maybe try to talk calmly to the kid. Tell her to that you want her to calm down. Tell her, that this is not the way to get what she wants. Tell her you can work out a deal, but first she needs to stop. Slowly and calmly, face to face. Take as much time as the process needs and do not try to rush it. Especially not for non-family members.
Also, think about whether raising ones voice and being loud is a behaviour the kid developed on his own or if he is mirroring some person (grown up or other kid) around him.
Ultimately, do not try to solve the alleged parenting problem, but the actual people problem. As others said, try to explain that this is a phase, that will be over eventually, and that you are working on it. The way you describe it, your neighbours are most likely not parents themselves (That's at least what I hope for their potential children), and they might not understand what's going on, no matter how well you explain it to them. In that case you'll need to develop some resilience against their complaints. Usually relations do not sour beyond repair though and get better, once the problem (by their definition) is gone. If your neighbours do stay mad at you after that ... well, I guess there is nothing you could have done in the first place. Either way, do not worry too much about your reputation with those people. Your family and your relation to your children is way more important.