You just interviewed someone for a job in your department. You liked him, so you offered him the job. He didn't immediately make a decision, so you say this to give him some more time to think about it.
Think it over and let me know in the next week or so.
To "think it over" means to think about something that you have to make a decision on. In the example above, the job applicant has to "think over" whether to take the job.
This expression almost always uses "it". If you don't use "it", the word order changes:
Have you thought over my offer?
A more formal word that means something similar to "think it over" is "consider". You would write this if you were offering someone a job in writing:
Please consider our offer and let us know what your decision is by next Thursday, March 10th.
"Let me know" means "tell me". But "let me know" is a more polite and friendly expression. It's very common to hear in an English-speaking workplace:
You can also use it with friends:
Let me know when you're finished with the computer.
This is an expression for talking about time loosely. "The next week or so" means "about the next week". It could be in 5 days, in 10 days, etc. Here's an example from the "Questions" page on this site:
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