The molecule in question has a 6-carbon cyclohexane frame with one carboxylic acid group, a hydroxyl group and two amino groups (the actual drug called Tamiflu is a phosphate salt). The starting material, bicycliclactone 1 already has several components in place but is achiral while the target molecule has 3 stereocenters.
Thioester 4 is formed by application of a strong base (KHMDS) and PhSSO2Ph. Grieco elimination to diene 5 is then induced by mCPBA. This compound is almost identical to the diene in Corey's version which his lab synthesised in about 7 steps from butadiene and acrylic acid. Trost claims his version is the shortest yet from a commercially available compound but Coreys starting materials are more easily recognized as simple raw materials you could buy everywhere.
The final steps are ring-opening of the aziridine with 3-pentanol and boron trifluoride, Acylation of the amino group and its deprotection with TBAF and finally removal of the phthalimide group with hydrazine.