If we change the sentence to 'Tom should have to attend the meeting.', it means that 'Tom should be required to attend the meeting.' Maybe the rest of us all have to attend the meeting, and we think that Tom also should (have to)(be required to) attend the meeting.
So in your sentence the meaning is 'I should (have to)(be required to) attend the marriage.'
As I said, it is a strange sentence. Perhaps the person who wrote this sentence meant 'I should attend the marriage.' or 'I have to attend the marriage.' (The last two sentences have meanings that are different from each other and different from the sentence you asked about.
Finally, sometimes "should" is used for the future, almost like 'will', but I don't think that is what it means here. I don't recommend using "should" to mean 'will'.