The phrase means "we will speak with each other again". "At snakkes ved"/"at tales ved" is just a kind of idiomatic verb structure. But really, "vi snakkes ved"/"vi tales ved" is almost only used as a conversation closer between two persons and almost exclusively in that form.
Vi snakkes ved i aften - we will speak to each other again this evening
Vi taltes ved senere - we spoke to each other again later on
De snakkedes ved næste dag - they spoke to each other again the next day
Freja og Thor kom aldrig til at tales ved den weekend - Freja and Thor never got to speak to each other again that weekend