Avoid thinking that "You just insulted me." Instead, think about it like this "they opened their mouth, and some sounds came out."
This is consistent with the Stoic practice of focusing on impressions without adding judgments. Impressions are not subject to will and it is incontrovertible that someone opened their mouth and made sounds. Judgments, such as the idea that one has been insulted, are willed. One can choose not to be insulted.
Or even more accurately, say "I felt insulted".
A person cannot choose to hurt your feelings. At most, he can say something hurtful and hope you are unwise enough to find such pettiness offensive.
But he cannot force a person to be upset by his words alone. You must become upset for him, so do not say "he upset me" for this is not accurate - say "I became upset by his words". Thinking in this more rational way, you immediately recognise that is by your own choice that you were upset, and by working on your choices you can avoid upset.
But if you say "he has the power to hurt me, I am powerless!" then you do nothing but weep and moan.
This can be a useful mindset to check your initial emotional response, but make sure you don't blind yourself to your aggressor's humanity and your fellowship with him.
Either the things he said are accurate and you can use them to gratefully improve yourself, accurate and meaningless and therefore meant only to provoke you, or inaccurate and the person is suffering from mistaken perceptions and therefore deserves pity rather than resentment.
Responding with gratitude, in any case, is warranted because he's challenging you to be a better person, either by giving you an opportunity to practice patience and to learn not to attach your virtue/happiness to the opinions of others or by actually offering you constructive criticism by which you can become a better human being.
My initial thought would be "he/she probably had a bad thing going on in his life". Usually, when my friends or even anyone are on edge, they say something bad about me I don't get mad at them. I just ask them if everything is alright. This question usually defuses any situation and makes them open up to me, but remember - you need to ask it in a sincere way.
I like 'Marcus' response to insults, to respond with kindness. I recall times I was mean to others and they responded with kindness, those times still haunt me, whereas if they had responded with anger, it would not have meant much.
The quote that comes to mind with insults is Hurt People hurt people. Someone is throwing insults at me, they are coming from an emotional state, trying to hurt me, while anyone who is in a healthy state would not be inclined to do that. I would only take an insult more seriously if it came from someone that's close to me in my life, because then I have to reflect is the insult a reflection of actions I have taken or possibly a reflection of my personality that I am not aware of and should be examined more closely.