It's probably firmly clipped together as is the modern design for items like this.

You may need to run a suitable object into the join between the two halves to separate them, this is common with many tablets, notebooks etc.

Generally it is a plastic tool like these (scroll down to "case open pry tool")

You may need to force the tool in to the join quite firmly and then slide it along to separate the case.

Just beware that often items like this are never designed to be repaired or opened, it's unlikely with this remote but with things like AC adapters "wall-warts" they also glue the join and there are extremely difficult to break apart and usually also damage the case - AND if you are not careful you can easily cut yourself.

There have also been a few designs of Remote control so without knowing your specific one it's hard to give any more advice.

And even once you get it open, it may still not be repairable. If it uses a rubber conductive pad like many TV remove and some older telephone keypads, then you can often clean the carbon pad and PCB traces and that is all that is required. I had to do this with my freeview remote recently, I was really surprised how dirty and sticky it was inside, (probably from hands using the remote while eating dinner!!)

But if it has little individual buttons or the domed 'clicky' contacts, then you might be out of luck to repair it.

Hopefully it will be a conductive rubber pad because that is the easiest and most reliable to fix and probably won't need anything replaced.

If you get it open, please post a couple of images.

Hope this helps

Steve
NZ