I'm curious to see how you made the text file in the first place, as there is probably a better way to store/extract the data you're after. However, to address your specific question I propose two options. If you can provide more detail of what the "other text" is, then this answer may be improved.
Option 1
Use split on '/` and grab the last element.
'/volume1/Media Library/Movies/Hotel Transylvania 2 (2015).mp4'.Split('/')[-1]
Hotel Transylvania 2 (2015).mp4
or
('/volume1/Media Library/Movies/Hotel Transylvania 2 (2015).mp4' -split '/')[-1]
Hotel Transylvania 2 (2015).mp4
Option 2
Use a regex pattern to extract the filename.
'/volume1/Media Library/Movies/Hotel Transylvania 2 (2015).mp4' -replace '^.+/(?=.+)'
Hotel Transylvania 2 (2015).mp4
To apply any of these in a loop you can use the Line property of the returned match.
Get-Content .\Movies.txt | Select-String volume | foreach {$_.line.Split('/')[-1]}
However the switch statement can both read the file and process line by line.
Switch -Regex -File (".\Movies.txt")
{
'volume' {$_.Split('/')[-1]}
}
Poltergeist (2015).mp4
Goosebumps (2015).mp4
Hotel Transylvania 2 (2015).mp4