Does anyone have any tips on understanding another person's reasoning, or when to give up? This is provided you're talking with them face to face.
For example today I was working in a group and one person had made a slide that was completely wacky and didn't make any sense. I asked him (politely) several times to explain, but each time it was gibberish. I never did understand it but since no one else could understand, we deleted the slide. His English was poor.
Does anyone have any suggestions? Especially when people are trying to communicate who don't speak the same first language, I find it can be a barrier when they just repeat the same broken English that doesn't make sense. Maybe asking "could you explain it in different words" would help, but that sounds a bit condescending in the work place.
You could try changing the question. Instead of telling them you don't understand what it means or asking them to explain it you could focus on a particular part and ask "What is this?", "Why is that thing important?", "How are this and that related", "Why is this an abomination from the depths of the early internet in neon colors and blinking?", etc.
A classic technique is to paraphrase back to them what you think it means. "So to me this slide says that the sky is purple and water is dry. Is that correct?". Then they can either confirm your understanding is correct or provide corrections if your understanding isn't correct.
These two can work really well together. Ask some small simple focused questions and then paraphrase your understanding of what you've covered so far.
As someone who has worked in a customer service role with plenty of clients whose English was at times very hard to understand I'd do the following:
If you don't really need to understand, let it slide. I've seen plenty of situations where both people are not understanding each other and the one trying to clarify is clearly just wanting to situation to end to avoid further awkwardness, if you get that vibe be gracious to them.
If you do need to know, depending on how poor their English skills are, they could be flustered and that's contributing to the problem. Wait until you get a moment in private with them and ask them then. It'll remove the pressure and you'll be able to work with each other to get to a mutual understanding.
As others mentioned, repeat back what you understood and ask for confirmation. If they say yes and then add some further clarification that you didn't understand, leave it at that as you got the gist and that should be fine. Once again, wait until you can speak privately if it's critical that you understand exactly what they're saying.
Similar to the first point, sometimes they will answer in the affirmative to your clarification questions just to end it. If you suspect this, speak with them in private to confirm or otherwise if you can, just let it go.
I assume that this slide was created with the intention of expressing a piece of information, and if you couldn't understand it that was a failure.
Your response depends on what your role is in relation to this person and how important that slide was. As a supervisor/project lead/etc I would ask the person to explain what they were trying to convey. If you still could not understand a good way to approach would be "I do not think your message is being conveyed clearly, can you suggest another way to put it" rather than "use different words". As you said is not the best to use in a business situation.
From there it may be time to reevaluate this persons role on the team. Perhaps doing a job where they need to express ideas in a language they do not know is not suited for them.