The class feature description has everything you need to know.
The feature description for Giant Stature is complete - you don’t have to look elsewhere to learn its effects:
While raging, you gain the following benefits:
Giant Stature. Your reach increases by 5 feet, and if you are smaller than Large, you become Large, along with anything you are wearing. If there isn’t enough room for you to increase your size, your size doesn’t change.
This is everything you need to know to play this character using this feature. You don’t have to look elsewhere to learn how this feature works. Thus, you get no extra damage, since extra damage is not listed as one of "the following benefits".
You don't need the Monster Manual or Dungeon Master's Guide to understand your class features.
The Dungeon Master's Guide contains guidance for the DM can use to create new monster stat blocks:
If you want a full monster stat block, use the following method to create your new monster.
The introduction to the Monster Manual explains all the components of a monster’s stat block. Familiarize yourself with that material before you begin. In the course of creating your monster, if you find yourself unable to make a decision, let the examples in the Monster Manual guide you.
Once you have a monster concept in mind, follow the steps below.
The instructions for creating a monster include the following guidance for bigger monsters, which is where the notion of oversized weapons comes from:
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it’s Huge, and quadruple the weapon dice if it’s Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3d12 slashing damage (plus its Strength bonus), instead of the normal 1d12.
This is guidance for the DM to use to homebrew a monster. This has nothing to do with player character class features. This section isn't even giving "rules" about anything. This is telling the DM, "If you want to create a monster, here's some help".
For more in-depth commentary on the so called "oversized weapons rule", see my answer to the question, How does the Enlarge/Reduce spell interact with the Oversized Weapons rule when you Enlarge a Medium size creature with a medium weapon to Large size?.
Answer adapted from this answer to the same question about a different class's feature.