I know as a DOTA player some of this is going to be elementary to you, but I'd rather go with the full explanation than give you a two-sentence write-off answer.
Put simply, grouping gives you a greater presence in your position on the map. This means that objectives (towers, inhibitors, neutral monsters, etc.) go down faster, and it's a far greater risk for enemies to engage with you unless they have similar numbers available. If your enemy decides not to commit their entire team to defend against your entire team, they're gambling on their ability to hold you off with inferior numbers while other team members attempt to secure other objectives. In addition, as the game goes on and as death timers get longer and players start upgrading their boots and getting more damage, they can respond more quickly and effectively to threats from groups of one or two players and remove them from the game for a minute at a time--more than enough to press a numbers advantage further.
Grouping defensively gives you the same benefit for the opposite reason. If the enemy is trying to take dragon, yes you can push your lanes and farm up, but that dragon buff, especially early in the game, is worth waaaay more than a couple minions or some tower damage. If you group near dragon and keep vision on it as the enemy team tries to take it, you can corner them in the dragon pit and fight them after they're weakened, or separate the team as they trickle in to contribute damage--similarly for Baron, except that Baron is an even riskier objective to take under pressure until high-damage carries start getting their fifth or sixth items. Grouping to defend towers, especially inhibitor and nexus turrets, provides additional threat to an area that the enemy team is already wary to approach.
Of course, grouping isn't always the correct strategy, and it depends a lot on your team composition and what the enemies are doing. Do you have a lot of area-effect damage and crowd control abilities? Then you gain the most from fighting against a lot of people at once. Hitting one person with Orianna's ultimate to secure a kill is good, but it's even more effective when you hit 4-5 people at once and bunch them up close together, leaving them perfectly positioned for other area-effect abilities. Having characters like this on your team will vastly increase your ability to win a teamfight--and as you said, take a very strong lead or even win if the game has gone late enough.
On the other hand, characters like Tryndamere and Nasus are often at their best while split-pushing. Each of them has a skillset that basically just lets them hit towers (or one enemy character at a time) really, really hard. This can be effective if Nasus has been farming his Q stacks and Tryndamere has a few attack damage and attack speed items, but this still leaves the rest of their team in a dangerous position: what if the enemy team decides to push mid as a group of 5? They're at a disadvantage, and if the enemy has strong tower-diving capabilities with a strong tank (or characters with pulls like Blitzcrank or Thresh), that could end disastrously for the split-pusher's team--and then the only thing stopping the enemy team from taking the same advantage is one guy who may or may not be as effective at taking towers as a 5-man team, and who is probably not going to be able to fend off the 2-3 players who will inevitably return--at full health--to defend their towers after winning the fight. If your team has strong peel to prevent tower dives, stall fights, and clear minion waves to keep the enemy team away, this becomes less of an issue--and it's potentially a strong reason to pick one of the above characters into a team with a Ziggs or Viktor who can clear minion waves from relative safety, or a Kayle or Braum who can provide a disincentive for the enemy team to dive under your towers to press the numbers advantage.
Timing also matters a lot. At level 1, grouping is almost completely useless except to invade the enemy jungle to secure jungle creeps (e.g. denying a mana-hungry jungler their blue buff) or cheese an early kill--and even if you succeed in those goals, you will often have lost time in lane where your opponents are gaining a crucial early farm and experience advantage. But if your bot lane duo hit level 2 before the enemy team and annihilated them, it might be advantageous to pull the mid laner and jungler in to take an early dragon at a time when the enemy jungler will be hesitant to contest it.
Ultimately grouping is about a numbers game. One team decides they want to take a tower or an objective, so they bring team members together to get it more effectively, leaving the enemy team with three options:
Fight at a disadvantage
Avoid the fight and permit the loss of an objective
Group themselves to even the odds
And that third option is how teamfights start. At lower levels of play teamfights can simply happen because players believe it's the natural next step in the progression of play, but more often as you climb the ranks it's because one team wants very strongly to take an advantage somewhere and their opponents want to stop them.
Grouping is absolutely a risk as death timers crawl higher, but that has to be weighed against the risk of leaving your team at a disadvantage if your opponents force your team to fight, because that can be equally disastrous. This doesn't always mean split-pushing is a worse choice, but (just like grouping) it has to be a decision made with consideration for your team composition, the enemy's team composition, the current state of the game, what objective(s) are being fought over, and whether you feel you can take something of greater value than what your opponents are getting out of it.
