The move was introduced in X&Y, i.e. Gen VI, so it's not as if it was an old move that got a boost, like Defog.
Sticky Web slows your opponent down by 1 stage, meaning it's perfect for Pokemon that have poor/ mediocre speed, like Mega-Mawile, Azumarill and Bisharp, which have risen in popularity in Gen VI, who can all have a chance of outspeeding some Pokemon they otherwise could not. Even for faster Pokemon, it's still great because it allows your team to out speed an opponent's Pokemon that might otherwise be at a speed tie.
It also makes Choice Scarf useless, as the web cancels out the speed gained from Choice Scarf, so common scarfers like Zoroark lose their function.
As for Galvantula's popularity, it's because it's fast and has nice offensive presence, as well as a really great ability in Compoundeyes, so it can utilise Thunder. Additionally, it trashes anything with Magic Bounce - Xatu, Espeon and Mega-Absol, in addition to having a really nice typing meaning only 2 weaknesses. There are other Pokemon that have Sticky Web that haven't risen in popularity, e.g. Ariados, because they don't have any other criteria to boost their usage.