In the given example, 再 is the "future again", i.e., future tense:
While 又 (yòu) is used for "again" in the past, 再 (zài) is used for "again" in the future.
Expressing "again" in the future with "zai", Chinese Grammar Wiki; see also Comparing "zai" and "you"
(Putting aside other grammar structures, such as 先...再....)
However, the 在 is short for 正在 which expresses actions in progress, i.e., present tense (or at least, continuity within the specified time period).
在 (zài) and 正在 (zhèngzài) can be used as auxiliary verbs to express that an action is ongoing or in progress.
Expressing actions in progress with "zai", Chinese Grammar Wiki
So 我再看电影 feels incomplete because you'd expect more context, e.g., a specified time period. Without 再 it could be 我又看电影了 ["I'm again watching a movie"], or with 再 we might say:
I will not watch a movie again.
I will go watch a movie again tomorrow.