First in the Object Editor you would give your unit an inventory at the top where it says "Abilities".
Then create your trigger and after it says "Unit - Create 1 (your unit)" add Set DropUnit for Last created unit. Then whenever you want the item dropped on your unit, you would do the following trigger. But just to get this right, you said that once a unit finishes training, you wanted the item dropped on the unit. So:
Events
Unit - A unit finishes construction
Conditions
Unit - Trained unit equal to (your unit)
Actions
Unit - Create 1 DropItemDummy
Unit - Set expiration timer on Last created unit to 1.00 seconds
Item - Create (your item) at position of DropItemDummy
Set ItemDropped to Last dropped item
Unit - Order DropItemDummy to right-click ItemDropped
Unit - Order DropItemDummy to drop ItemDropped on (your unit)
Have fun!
Edit: I had started writing this before someone else had posted... but I had to go somewhere so I couldn't finish. Anyway, I hope that this can still help you.