It's informal. It's just a way of asking for someone's opinion.

A photographer "takes" a picture. In the movies, a "take" is a piece of film recorded without stopping. In the expression "what's your take," I <em>think</em> the analogy is choosing a camera angle.

In other words, "what's your take on this" is another way of saying "what's your point of view?" In both cases, idea is that your opinion is based on your particular area of expertise.

"Pete, from engineering: what's your take on this proposal?"

"That roller has to be awfully rigid. I'm not sure our supplier can meet the spec."

"Ellen, from manufacturing, your take, please?"

"I'm not sure it's manufacturable. Please explain, again--how does the assembler get into that place to install that bolt?"

"Lewis, from marketing, your take, please?"

"I'm concerned about the schedule. Remember, we want to have it ready to show at the trade show in August."

I think there is also a sense that a "take" is a short or summarized opinion.

Asking for someone’s take is just another way of asking for someone’s opinion/point of view. Another way of asking “what do you think of this/that?”

I would use it when talking about a specific thing. For example, if you are having a conversation about a political situation, the environment, or anything else like that. It is basically just asking the other person’s opinion on a specific subject. Basically a “what do you think about that?” Or “what’s your opinion on that?”.