One way that DVD decoding programs extract the subtitles is into a pair of files: IDX and SUB. These are popularly known as the VobSub subtitle format.
The SUB file is a binary file containing the bitmaps of the subtitle texts. The IDX file is a text file with a list of entries, one for each subtitle. Each entry has the time when it should be displayed in the video and a value indicating the offset into the SUB file where the subtitle text bitmap can be found.
Video players like VLC can directly read and place subtitles in the IDX-SUB format if their files have the same name as the video file and are placed in the same directory.