How does the entropic force arise within the hydrophobic effect?
Whenever a non-polar substance is put into water, water molecules will organise themselves around it in a cage-like formation. The molecules exposed to the non-polar substance at any given time will orient themselves to form as many hydrogen bonds with the rest of the solution as possible.
(Image from here)

The way I understand it intuitively is that these water molecules are restricted in the orientations they can assume, becoming more ordered to maximise hydrogen bonding, so have an overall lower entropy than if they were free to tumble like in the bulk solution. Similarly, the aggregation of non-polar molecules also allows more free movement within the non-polar 'blob'. If they were distributed evenly, there would be a higher surface area and therefore amount of 'cage' water molecules and non-polar solutes would be more restricted in their movement (trapped in a polar environment). This is the entropic factor that drives the hydrophobic effect.
It is also favourable enthalpically; hydrogen-bonding interactions and non-polar interactions are maximised respectively.
These two approaches demonstrate why it is thermodynamically favourable for non-polar substances to aggregate, not disperse. To clarify, it is still thermodynamically unfavourable to mix the two in the first place, that's why they separate; this is just the better option if they were to mix.
It is also worth noting that oil 'blobs' in water tend to be very round, due to the fact that a sphere is the shape with the smallest surface area to volume ratio. This supports the above reasoning, minimising the number of 'cage' water molecules whilst maximising movement of both water and the hydrophobe.