Please explain the correct usage of "rainy". Is the sentence "It
became rainy." correct? (Without anything following the adjective)

I'm no grammarian (nor an official researcher) but I think the
sentence "it became rainy" is fine, and the reason it's fine doesn't
really have anything to do with any special quality of the adjective
"rainy." Rather, English grammar permits the use of what are called
"predicate adjectives"--adjectives used after certain linking verbs
such as, in your example, "became."

Try "it began to rain."  I'm not sure of whether "it became rainy" is
correct grammar, but the phrase "it began to rain" or event "it
started raining" or if your talking about just a little bit of rain,
then "it began to drizzle."

I think it's slightly colloquial but perfectly correct.  You might
say, for instance, "It became rainy towards the end of the
day/holiday" which probably means the same as "it started to rain on
and off towards the end of the day" or "we had several showery days
towards the end of the holiday" but is shorter.  You probably wouldn't
use the phrase in a formal written context such as an academic paper.