The subtitles are a text document you download, basically. They are usually in the .srt or .ass format. You can right-click on these files and select most text editing programs to open them in.
Here is an example from an .srt file opened using a basic text editor:
1
00:00:08,307 --> 00:00:19,535
2
00:00:19,535 --> 00:00:21,835
(うめ先生)おはよう
3
00:00:39,789 --> 00:00:42,089
(宮子)いただきます
4
00:01:02,795 --> 00:01:04,797
(沙英)あッ…
5
00:01:04,797 --> 00:01:06,783
・(ヒロ)沙英~?
6
00:01:06,783 --> 00:01:09,083
ああ ちょっと待って
7
00:01:16,809 --> 00:01:18,861
(食器の割れる音)
8
00:01:18,861 --> 00:01:21,781
ゆのっち~ どうした?
9
00:01:21,781 --> 00:01:25,451
(ゆの)宮ちゃん ゴメン
学校 先に行って
10
00:01:25,451 --> 00:01:27,751
えッ? ありゃ~
11
00:01:29,789 --> 00:01:33,126
どうしたの?
大丈夫?
12
00:01:33,126 --> 00:01:36,126
沙英さん ヒロさん すみません
13
00:01:37,130 --> 00:01:39,730
これは派手にやったね
14
00:01:41,801 --> 00:01:44,137
ぺったん ぺったん
15
00:01:44,137 --> 00:01:46,456
よしっと こういうこと
16
00:01:46,456 --> 00:01:49,776
急いでるときに限って
やっちゃうんだよね
The media player/SRS program uses the timing data for each line/section to display the related text on the screen in sync with the sound.
.srt files are probably the easiest option for simply opening with a text editor, as .ass files are more complex.