What does "repeat these 3 words back to me" test?

This is just a small part of an MME or Mini Mental Exam.
This is not a great way to assess a persons memory and if this is all they have done I would not put much stock in the results.
And a MME is not given just once it is repeated months later to determine how much of a decline there has been. So if this was the first and only time she was given this "test" I would put even less value to it in the matter of results.
Funny thing is when they gave my Husband 3 words to remember they were something like Denim, Cotton, Apple. He did not recall all of them and I said later if they has said Screwdriver, Hammer, Race Car he would have been able to recall the words.

It's an assessment of short term memory.

I can't answer this.

They did this with my Mom.

I liked what Veronica asked "I am curious, were you with her and did you remember the words? If yes, how easy or difficult was it to remember?"

I was with Mom when she was tested by a neurologist. There were tests he gave Mom I had problems with. One was looking at a picture and then drawing it from memory. Mom did great, don't think I could have.

Toothill is correct it is part of a simple cognitive test. If you are interested you can find the entire test on the web. This does not give a complete diagnosis it just helps healthcare professionals get an idea of their patients mental abilities.
As an inpatient I was often asked where I was, the name of the hospital,city etc.

Short term memory recall. I have had it during employment interviews too. It is one of many measurements used to check for cognitive decline. I am curious, were you with her and did you remember the words? If yes, how easy or difficult was it to remember?

This test can be very important when you think about new actions that a person may need to take. If your Mum has a new prescription will she remember to take it? Even if she can remember to take the Rx she has been on for years, she may not remember the new one. Will she remember to do her exercises once she is back home? Again, short term memory recall in at play here.

If Mum does not remember having a conversation or phone call, this can be indicative of a problem. Often people post here about family members complaining that no one visits, when the family knows there are regular visitors. Yup, could be short term memory issues.

I went through an incredibly stressful time 5 years ago. At that time my short term recall was terrible. I had to use notes, which I did remember to check, in order to function .