Everything that drops lands in your bag. Gear goes on your character from the Equipment screen; what you do not need is broken into materials.
1The bag
MENU > INVENTORY: every item is a tile (1). FILTERS (2) narrows the grid; DISMANTLING (3) opens the breaker.
2Read an item
Click a tile to read the piece. LOCK (1) keeps it out of Dismantle All and Auto-optimize; DISMANTLE (2) breaks this one piece into materials.
3The Equipment screen
MENU > EQUIPMENT: your character's gear slots (1), each with its gem, rune and scroll sockets. AUTO-OPTIMIZE (2) wears the strongest legal pieces from your bag; BUILDS (3) saves a set to come back to.
4Equip and unequip
Click a slot: the piece in it is under CURRENTLY SLOTTED (1) - click it to take it off - and everything else in your bag that fits is under AVAILABLE (2); click one to wear it instead.
5The ILLEGAL mark
A red ILLEGAL mark (1) means a worn piece no longer fits - the reason is next to it: LV 30 is a piece above your level, MAGE ONLY a weapon of another class, WRONG SLOT a piece in the wrong place. It stays on until you swap it; Auto-optimize never picks an illegal piece.
6Dismantling
DISMANTLING breaks gear into Scrap and materials, four ways (1): everything the current filters show, every piece of the grades you pick, every piece of the tiers you pick, or pieces you mark one by one with the hammer cursor. Locked pieces are never touched.
Screenshots are taken from the live game by tools/tutorial_shots.py (dfmc-client) and regenerated after interface changes. The rules behind each screen: system-equipment, item, reliquary.