Why is a drop of chloroform stretched/pulled apart on a water surface?
As you pointed out, the surface tension of water-air interface is greater than the surface tension of chloroform-air interface. At $25~^\circ\mathrm{C}$, the surface tension of water-air interface is $71.99\pm0.05~\mathrm{mNm^{-1}}$ and that of chloroform-air interface is $26.67~\mathrm{mNm^{-1}}$ (source).
Now, let's understand why a drop of chloroform placed over water gets puller apart forming a layer. Surface tension is the force per unit length. More the surface tension, more is the force per unit length and vice versa. Surface tension is similar to tension in a stretched rubber sheet. As you place a drop of chloroform on water, the system looks as shown in the following diagram:

As we seen earlier, surface tension of water-air interface is greater than that of chloroform-air interface. So in the boundary separating chloroform and water there exists an imbalance of force. The force vectors are the red arrows in the diagram above and their strength is denoted by their length and weight (thickness).
Due to this, the drop of chloroform expands to form a layer on top of water. This can also be looked in the nature's tendency to lower energy. The surface energy of chloroform-air interface is lesser than that of water-air interface. So, a lower energy configuration is preferred.