What does the “15w40” stand for?

15W40 means that at cold temperatures the oil has the same viscosity as SAE 15W oil. At high temperatures it has the same viscosity as SAE 40 oil
The Society of Automotive Engineers (SAE) has established an arbitrary scale for the viscosity of motor oils. The scale ranges from 0 to 60.
The numbers from 0 to 25 have the letter W added. This means that they are “winter” viscosity, for use at lower temperatures.
The viscosity of a liquid is its resistance to flow. High intermolecular forces between the molecules cause a high viscosity.
As the liquid warms up, the added kinetic energy overcomes some of the attractive forces. The viscosity decreases. Hot molasses flows more readily than cold molasses.
A single grade oil like 15W or SAE 40 oil has a high viscosity when cold and a lower viscosity when hot. The first number 15W is the viscosity of the oil at cold temperature, and the second number 40 is the viscosity at 100 °C.
The 15W40 designation means that the oil is a multigrade oil. It has the viscosity of 15W when cold and the viscosity of SAE 40 when hot. This means that one type of oil works in all temperatures.
The diagram below shows the viscosity behaviour of 5W40 oil.

(from www.bimmerfest.com
At -30 °C, 5W40 has about the same viscosity as SAE 5W. SAE 40 would be too viscous to use.
As the engine heats up to 140 °C, the viscosity becomes about the same as that of SAE 40. SAE 5W would not be viscous enough at this temperature.
At all times, the multigrade oil has the correct viscosity for the conditions.