Thus, I think, the orgins are pretty clear - the apron was the item that denoted a working or servant girl; it covered an area which excited the interests of rogues and if they had their evil way there was a risk of pregnancy.
To say 'and now my apron strings won't tie' is perhaps rather more subtle than 'he got me with child,' and is probably more suitable for a female singer in a mixed audience than the blunt alternative, in that she can be as demure or as suggestive as she likes - either blushing demurely or winking lewdly as she delivers the line.
I've often associated such apron songs with the cuckoo's nest variety, where again an element of euphemism is used to suggest sexuality.