The linked question in the comments provides a very thorough explanation. To answer your question more simply though, the meaning of should in this example implies that it was almost expected to begin raining, given the circumstances or situation.
It is also used in a similar way to describe behaviour that someone hopes or imagines is going to happen. (Or, conversely, when it was not expected or hoped to happen)
My boss insisted that the work should have begun much earlier.
For the cold weather we're having, it's unusual that the trees should have blossomed in March.
It is a more common construction in British English, including everyday speaking, as far as I know. In American English, the "should" can often be omitted and the same meaning still implied. For example:
My boss insisted that the work begin much earlier (American English)
My boss insisted that the work should begin much earlier. (British English)
This construction is sometimes referred to as a pseudo-Subjunctive. For example at this website here: http://www.englishclub.com/grammar/verbs-modal-should.htm
The linked answer goes into a lot more detail, but I hoped to give a more concise answer here.