What is the work done by a force acting on a deformable body?

If the forces were constant as the deformations increase from zero to the fully deformed state, then the work done at each force would be

$$W=Fdelta$$

However, the text is making the assumption that each force is increasing from zero to 100% while at the same time the deformation is increasing from zero to 100%. Therefore, the work done at each force is the average force times the deformation.

$$W=Fdelta{}/2$$

The above is true for elastic deformation, where the ratio of force and displacement is constant.

For plastic deformation, the average force will not be half of the total force. In this case, a material stress vs. strain curve would be needed to calculate the average force.

The text also mentions that we assume the force application is gradual. This simplifies the situation by removing inertia considerations.

Addition below to address your question clarification...

A force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. Notice that the definition of work doesn't say that the displacement was caused by the force.

The reason the works at each point can be summed up to get the total work done is because each force is ramped from zero to its maximum value during the same time period. This is what is meant in the text about a fraction of each load being applied at the same time. This load application process ensures that each point moves only from zero deflection to its maximum deflection.

Assume instead that $F_1$ is fully applied first and then $F_2$ is applied. In this case, the length of the path traveled by point 2 would likely be greater, and the work done at point 2 would be correspondingly greater.

In other words, the work done on a body is path dependent. In a similar way, the work done lifting an object directly up one flight of stairs is less than the total work done lifting it up two flights and then back down one flight.