"The team with the pushing lanes is advantaged early game." This phrase is unanimously understood to be true by the coaches, analysts, and casters that study the game on a regular basis, but its truth is a bit less self-evident to those of us that play the game on a non-professional level. Lane control is one of the most poorly understood concepts in the game outside of the top tiers of play. Outside of "if we shove bottom lane, we'll get level 2 first", the value of having a shoving lane is lost on many, a fact that's not surprising considering the lack of coordination outside of competitive games.
It's a simple and intuitive concept once it's explained, however. The simple fact is that having a shoving lane grants you the ability to roam more easily. True, you can roam whenever your heart demands it, but doing so before wiping out the enemy creep wave is a recipe for taking significantly more tower damage than you need to, all while falling behind in XP if your roam proves fruitless. More to the point, the enemy laner will also have an advantage in matching said roam, as if they have shoved your wave into the turret they have easier access to the river which tends to be the nexus of most early game action in League of Legends.
Furthermore, a pushed lane grants more safety and flexibility to your teammates. If you're a mid laner, you better believe your teammates are going to play under their turrets if you are as well so as not to risk death via Realm Warp. If your bottom or top lane isn't pushing you can forget about contesting the enemy buffs if you're a jungler. Without the support of your teammates, such a move is a death sentence. Pushing the top lane means that the enemy top laner will have far fewer opportunities to teleport into fights in other lanes undetected, as you're the one who will have access to the brush.
All this to say, there's always an advantage to shoving out your lane. There are risks as well, especially if you stick around to harass under turret, but with proper warding and attention that risk is a manageable one. The advantages far outway the risks, however, as pushing out your lanes grants you an excess of the most important currency in League of Legends: control. Remember that the team with the most options tends to be the team in the driver's seat and play accordingly.