Learn Swedish: Vi träffades på samma ställe igen.

The -s makes it a reciprocal verb, meaning the action is done to each other. Vi träffade would mean "we met (someone else)", but vi träffades means "we met each other".

Why is it samma ställe instead of samma stället?

The word samma (same) is a special adjective. It is almost always followed by a noun in the indefinite form, even though the English translation uses "the" ("the same spot"). So you say samma ställe, never samma stället.

Why do we use the preposition på here for "at"?

In Swedish, you typically use på with the word ställe (place or spot) when talking about a location. You can think of it as being "on" a particular spot.

Could I use plats instead of ställe here?

Yes, på samma plats works perfectly and means exactly the same thing. Ställe is just slightly more casual and very common in everyday spoken Swedish.