Translation:Grandpa hugs the groom and says congratulations.
Yes, that's right. In written language you can of course leave out the "että" but only when you put the quote between quotation marks. But somehow this Finnish sentence still looks bit weird for me and I'm not exactly sure why... maybe because my mind just all the time tries to put it in the quotation marks in written language.
The two languages are not one-to-one equivalents; they have different structural requirements. In Finnish the sentence needs että to tie everything together, where the comparable English sentence doesn't. It's just one of those language rules we need to get used to. I expect Finns feel the same way when learning English, wondering where all the ettäs have gone!