I have heard of many different people watering seedlings with dilute H2O2 becuase it does carry some oxygen down to the root hairs.
Me, I would rather use a very open, coarse, soiless mix that let's beaucoup AIR get to the root hairs.
But peroxide might discourage damping off, at least a little.
I like the theory that H2O2 disourages mold and bacteria while soaking seeds overnight. And MAYBE in some cases it helps break dormancy by oxidizing some germination inhibitor, or helping to strip something off the seed coat.
Several people said they saw older seeds germinating faster or with a higher % germination.