“Here - let me take a look.”
Your coworker is trying to change the color of some text on a spreadsheet document on his computer, but can't figure out how to do it. You think you know how, so you say this.
Here - let me take a look.
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People often say "here" before offering to help or take responsibility for something. A few more examples:
Here - let me do that
Here - I'll hold this while you tie it up.
Of course, you can also use "Here" in lots of other situations, like when you give something to a person:
Here, I brought you something from Singapore.
The word "look" is very general. It can mean to look for a long time, a short time, carefully, absent-mindedly, or in many other ways. "Take a look" is more specific. It means to look at something for a short period, usually for a specific purpose.
You can ask someone to take a look at something that needs to be fixed, or for something that's important for them to see. Here's another example from a science classroom. The teacher tells the class to look at something in the textbook:
When you're offering to do something to help someone, you say "Let me ___". For example, when your friend arrives at an airport and is carrying heavy bags, you can offer to help carry them by saying:
Let me help you with those.
When you use the phrase "Let me ___", it's common to end the sentence with "for you":
Let me look that up for you.