One reason is Gold math. If you're buying some starting item(s) and regen, you might end up with roughly 100 Gold left. Which is enough for 2 Branches, but not for anything else. Now you might argue that you could just save that gold for later, but that's a bad idea in general because the +2 to all stats give you a much greater benefit than getting your later item 3 creeps earlier (also see below). Similar calculations apply if you're left with 50 or 150 Gold.
Another important reason is that Branches are extremely cost efficient and very space inefficient. They use up an entire slot for a mere +3 total boost, but cost just 17 Gold per stat, which is tremendously low. At the start of the game, you have little Gold and much space in your inventory, so Branches are a perfect fit.
Lastly, they turn out really, really strong when you compare advantages to disadvantages. Assuming you're not building them into anything and just selling them later, they actually cost you just 25 Gold. However, getting a +1 in each stat not only increases your survivability by a small amount (but at the start of the game, this small absolute value turns out to be not that small in relation to your stats out of the box), but most importantly increases your damage, making it easier to last hit and deny. 1 damage might not seem much, but if it costs you 25 Gold in the long run, it's really worth it. Chances are, you'll get at least 1 last hit you would have missed without them, and 1 is already enough to pay them off. That's not counting extra regen and extra harass damage.
If you even build them into Magic Wand or Mekansm, you get items that are very efficient and useful well into mid game, while preserving the stat gain from the original branches and restoring inventory space. Those are especially useful if you're not steamrolling the enemy team, so getting Branches early can help you get some comeback options slightly cheaper.
What it boils down to is that they're so damn cheap Gold wise that the only real reason not to get them is space constraints, which you usually don't have early.