Let us call the 3 young couples, A and A', B and B', C and C'
The primes are the brides or ladies.
The banks of the river are the near bank (where they are at first) and the far bank (where they want to go).
1st passage: A and B ride for the far bank. Both men.
A' and B' are with C and C', so no jealousy here.
A is left at the far bank, while...
2nd passage: ...B rides alone back to near bank.
3rd passage: A' and B' ride for the far bank. Both ladies.
A' is left at the far bank with her husband A, while...
4th passage: ...B' rides alone back to near bank.
5th passage: B and C go for the far bank. Both men.
B' and C' are left on the near bank.
B is left at the far bank, with the couple A and A', while...
6th passage: ...C goes back alone to the near side.
At the near side now are B', C and C'. No jealousy. B' is not alone with C.
7th passage: C and C' go for the far bank.
B' is left alone at the near bank.
8th passage: B goes back alone to the near bank to fetch his bride.
9th passage: B and B' cross to the far bank, and all 3 couples are at the far bank now.