You have two methods of putting hydraulic fluid into a system: closed and open. Closed systems require a pressure system to keep the pressure up, open ones use a pressurized hydraulic that is expended. The open system was used in this situation to reduce the weight necessary to run a closed system.
As I understand it, there are fins that deploy in order to steer the rocket on the way back down (the first stage). They are really cool "Grid Fins" [1] that work even in supersonic flight. In particular, they are used to prevent the rocket from spinning, and to keep the rocket upright.
The rocket can also steer by "gimbaling" the motor (pointing it in different directions), but that has limitations; changing the rocket's orientation will also change its velocity (like balancing a broom on your hand).
In this case, the hydraulic fluid is used to rotate the grid fins and therefore keep the rocket upright. When the fluid ran out, the fins wouldn't work, which is why the rocket comes down tilted.
The hydraulic oil is not used as a 'fuel' to burn but to control the actuators for the grid fins. So if you lose control because you run out of hydraulic oil the last bit of the descent will be uncontrolled. So they almost made it with enough hydraulic oil, which means that next time the grid fins will work all the way down, which leaves a ton of other possibilities for mishap (too high a descent rate, rogue waves, barge being slightly at an angle and so on). Makes you wonder why the hydraulic circuit isn't closed, probably they figured an open loop system is more reliable than a closed one (half the gear and no need for a sump, which comes in handy when you have to deal with gravity coming from all directions).
Open loop is lighter than closed. No need for a closed loop if you have a set time of operation. I thought I heard they were also pumping the fuel through the system as well so I imagine this amounts to just a bit more propellant and another go.
The fins turn and are powered by hydraulic fluid. When it ran out, the fins stayed pointing in a certain direction, which meant the stage couldn't steer. Then it hit the ship.
