Unfortunately you do not say which country you are in and this is important information in determining your best course of action.
My background is that I have lead many Software development teams in the US and Europe and also lead hybrid onshore-offshore teams in Eastern Europe and Asia.
When suffering from stress, it can be difficult to deal with situations that would otherwise not cause you to skip a beat. It is also easy to give too much importance to events that have happened and to make it worse by over-reacting after the fact.
My experience is that in a large team/organization these kinds of things happen frequently enough. There will always be some team-member that is struggling either professionally or from challenges in their personal life.
My advice is as follows:
1) Don't make it worse by over-reacting. It's really not that big a thing to leave a standup.
2a) If you have a good relationship with your manager then you should communicate with him/her. Give them some insight into the challenges that you are coping with. You do not need to share every detail of your life but enough so that the manager can understand that you need a bit of support to get back on track. A good manager will give you reasonable accommodation. Every employee needs this at some time in their career and no manager wants to lose someone from the team because of temporary problems.
2b) If you do not have a good relationship with your manager and you feel that you can not share the struggles that you are dealing with, then you should simply say that you felt ill and needed to leave the enclosed space of the standup. Dealing with extreme stress and becoming overwhelmed is no less of a health issue than being overcome by nausea for example.
3) Do not feel the obligation to explain yourself to every team member. This will only draw attention to something that while it was a significant event for you personally, was probably not much more than a blip to the other team members.