Danielle Domae said:

Can you please advise what should I do?

Put the files to the right place ;-)

Really, from your description it simply seems that you are putting the files to a completely different place than where the actual project is located.
Check the project properties, there you can see the project folder location - put the translated SDLXLIFFs to the appropriate target language subfolder in that location and then you simply MUST see the translations after opening the files from within Studio project.

Of course, all that only if the SDLXLIFFs you received are actually really translated! Could it be that they are simply NOT translated?

Oh, only now I see that you mention XLIFF file, not SDLXLIFF files!
If they are just generic XLIFFs, not Studio-specific SDLXLIFFs, you won't be able to see them in Studio using this way...

Yes, I placed the files in the right folder since it shows me the path on the properties panel on Studio. And after I create a new project, the files are indeed translated. 

What you think I should do to substitute them if they are just XLIFF files? 

If they are generic XLIFFs and not Studio-specific SDLXLIFFs, then they simply cannot be used as direct drop-in replacements since Studio will not read them.

You would need to convert them to SDLXLIFFs... e.g. by creating new project - as part of the project preparation, your souce files (these XLIFFs) are converted to SDLXLIFFs.

If you want to use the "drop-in process" which you described, your supplier must send you SDLXLIFFs, not generic XLIIFFs.
And just to be clear - since I've seen translators doing many weird things in my career - a simple renaming from XLIFF to SDLXLIFF is not enough...