When you have a three-space extension on the third line, the attachment below is a tactic that you can use to connect them.
In the general case of two stones three spaces apart, if the opponent places a stone in between them, he probably intends to split.
As always, read carefully and don’t believe everything that your opponent asserts at face value
If your stones really are split apart, for the love of AlphaGo, don’t attach! In fact, don’t play against the splitting stones at all (this applies to any splitting situation). To defend your groups, lean on some other stones of your opponent, or sacrifice a part of them.
In your game, none of your stones were really threatened. Rather, it seemed like you were bending over backwards trying to defend some really thin strips of territory. Show some fighting spirit and counter-attack, or take sente elsewhere.