What should my reaction be when Mother feels someone is coming into her room in an assisted living facility during the night?

Your reaction to your mother should be "we will find out what's happening."

Then find out. It probably is a member of staff carrying out routine checks, so as a first step ask the facility about it.

Is your mother upset or worried by the visits?

In my Aunts AL it was a staff member checking on her. She was in her 90s so I see the reasoning. If this is what is happening and it scares Mom, I would talk to the Nurse in charge and ask if there is an alternative.

Have administration/staff check cameras to see who might be entering her room at night. Reassure her by really checking it out...not just telling her you are. Not all who are diagnosed with a memory issue just make things up. There could be some truth to what she is saying. Your job is to protect her and ease her anxiety. Not everything you hear from staff or other residents is truth. Just moved my mom 2 months ago because of abuse. And it wasn't because she told me something happened...I saw it for myself in the bruises on her arms and questioned staff. Thank God someone saw what happened and wasn't afraid to speak up for my mom.

It can be hard to know the difference between truth and delusions, when my grandmother was in AL (they called it a retirement home back then) she had a wandering man come an crawl into be with her. Do you know if the aides routinely check on them through the night?

Is this a frightening "visit"?
You can tell her that often staff will "peek" in to make sure everything is alright and she is comfortable.
If this "visitor" is frightening to her tell her that the staff is right there in the next room and that they will make sure that she is safe and nothing, no one will harm her.
It is possible that she hears noises outside her room and she may think someone is in her room. Other residents may wander at night and this might be what she is hearing. Or staff walking the hall, talking to other staff.
Reassure her that she is safe and no one will harm her if that is a concern.

I would begin by giving her assurances that you've reported this to the admin and they are investigating it. This is called a "therapeutic fib" and the purpose of it (if you are really certain this in fact is not happening) is to reduce her anxiety. You don't say if your mother has a diagnosis of dementia, ALZ, Parkinsons, etc. or a history of mental illness, but if you are her medical PoA and Medical Representative you may want to discuss this with her doctor, at which time he/she may recommend meds that can address this if it continues or worsens. Don't argue with your mom as it is very real to her.