Plug the card into the card reader and the card reader into the phone. Open a file manager (ES File Explorer, MyFiles, Total Commander - whatever your favorite one is), copy the files from the SD card and paste them to the phone. (With Total Commander in landscape mode you can open 2 windows, the card and the phone, and just drag&drop.)
From 4.4 om, the card is part of the file system - only apps with permissions to write to that folder can write to it. If you root the phone, you can run NextApp SDFix: KitKat Writable MicroSD (root) to change that back. (Restart after running it.)