Breeding is a pen, not a trade: put a pair in, the pen eats Ichor minute by minute, and when its bar fills an egg drops into your inventory. Pen 1 is yours from the first hatch; the other five are bought in order.
1Open the pens
Open the MENU and pick BREEDING. The plate is dark until breeding unlocks on floor 8 - the lock line under it says the floor.
2The pen grid
Six pens in a 2 x 3 grid. Pen 1 already holds a pair; its door is open and the parents, their grades and the progress bar show inside. Each closed door says what it costs to open - 12,000 Ichor for pen 2, 60,000 for pen 3, then Wraithgold - and they unlock in order.
3Ichor is the timer
Your Ichor sits in the header. A running pen burns it minute by minute and fills its bar; when the Ichor runs out the pen pauses and says so. Extra pens run in parallel - they do not make one egg come faster.
4Hatch the eggs
Eggs land in the INVENTORY. Click one to hatch it, or HATCH ALL to open every egg you can; the hatch level in the egg's corner says how far the creature has to climb before the egg opens.
Screenshots are taken from the live game by tools/tutorial_shots_b.py (dfmc-client) and regenerated after interface changes. The rules behind each screen: system-breeding, eggs, system-grade.