The phenomenon behind the nylon fiber's preference for the liquid/liquid interface is related to the interfacial tension and the surface energy of the fiber.
Interfacial tension is a measure of the force per unit length that acts along the boundary between two immiscible liquids, such as an ionic liquid and an organic solvent. It is caused by the imbalance of attractive forces between the molecules of each liquid. The interfacial tension tends to minimize the area of contact between the two liquids, forming a curved interface.
Surface energy is a measure of the excess energy at the surface of a solid or a liquid, compared to the bulk phase. It is also caused by the imbalance of attractive forces between the molecules at the surface and those in the interior. The surface energy tends to minimize the surface area of a solid or a liquid, making it more compact and spherical.
When a solid, such as a nylon fiber, is placed in contact with two immiscible liquids, it will experience different forces from each liquid. The force per unit length exerted by each liquid on the solid is proportional to the cosine of the contact angle between the solid and the liquid. The contact angle is determined by the balance of interfacial tensions between the solid/liquid, solid/gas, liquid/liquid, liquid/gas, and gas/gas interfaces.
The nylon fiber will tend to position itself at the liquid/liquid interface if the sum of the forces exerted by each liquid on the fiber is greater than the force exerted by air on the fiber. This can happen if each contact angle between the fiber and each liquid is acute, meaning that both liquids would wet the fiber. In this case, the fiber will reduce its surface energy by maximizing its contact area with both liquids, while minimizing its contact area with gas.
To understand this phenomenon better, you can search for concepts such as interfacial tension, surface energy, contact angle, wetting, and Young's equation. You can also find some information in these web pages: Liquid/liquid interfaces: a unique and advantageous environment to prepare and process thin films of complex materials and Synthesis of a Polymer: Nylon-6,6.