The principle of conservation of angular momentum says that angular momentum remains conserved unless an external torque acts on it. The net torque on a body is defined as: $$\vec{\tau\,}=\dfrac{\mathrm d\vec{L\,}}{\mathrm dt}$$ We can clearly see from this definition that since external torque on the body is zero, the angular momentum is going to remain constant. But the angular velocity is not, and that is what which changes with change in angular momentum, because: $$\vec{L\,} = I\vec{\omega\,}$$ For example, ice skaters when have their arms outstretched, their moment of inertia is high and so angular velocity is low, but if they draw in their arms, their moment of inertia decreases and correspondingly, without any external torque, their angular speed increases!

###Edit: The revision to your question has made it further interesting. Imagine that the rod is connected to a motor. Now, once the insect starts crawling towards the end, the moment of inertia of the entire system increases. According to our equations, the angular speed of the system should correspondingly decrease. But, we are told that it remains a constant $\omega$. This means that the motor has continuously apply a torque to keep the angular velocity constant! This is the external torque that we have find in the question, and it is responsible for the increasing angular momentum as well as kinetic energy of the system.

In the first case, there was no constraint keeping the angular velocity constant, unlike your second question.

The principle of conservation of angular momentum says that angular momentum remains conserved unless an external torque acts on it. The net torque on a body is defined as: $$\vec{\tau\,}=\dfrac{\mathrm d\vec{L\,}}{\mathrm dt}$$ We can clearly see from this definition that since external torque on the body is zero, the angular momentum is going to remain constant. But the angular velocity is not, and that is what which changes with change in angular momentum, because: $$\vec{L\,} = I\vec{\omega\,}$$ For example, ice skaters when have their arms outstretched, their moment of inertia is high and so angular velocity is low, but if they draw in their arms, their moment of inertia decreases and correspondingly, without any external torque, their angular speed increases!

Edit:

The revision to your question has made it further interesting. Imagine that the rod is connected to a motor. Now, once the insect starts crawling towards the end, the moment of inertia of the entire system increases. According to our equations, the angular speed of the system should correspondingly decrease. But, we are told that it remains a constant $\omega$. This means that the motor has continuously apply a torque to keep the angular velocity constant! This is the external torque that we have find in the question, and it is responsible for the increasing angular momentum as well as kinetic energy of the system.

In the first case, there was no constraint keeping the angular velocity constant, unlike your second question.

The principle of conservation of angular momentum says that angular momentum remains conserved unless an external torque acts on it. The net torque on a body is defined as: $$\vec{\tau\,}=\dfrac{\mathrm d\vec{L\,}}{\mathrm dt}$$ We can clearly see from this definition that since external torque on the body is zero, the angular momentum is going to remain constant. But the angular velocity is not, and that is what which changes with change in angular momentum, because: $$\vec{L\,} = I\vec{\omega\,}$$ For example, ice skaters when have their arms outstretched, their moment of inertia is high and so angular velocity is low, but if they draw in their arms, their moment of inertia decreases and correspondingly, without any external torque, their angular speed increases!

The principle of conservation of angular momentum says that angular momentum remains conserved unless an external torque acts on it. The net torque on a body is defined as: $$\vec{\tau\,}=\dfrac{\mathrm d\vec{L\,}}{\mathrm dt}$$ We can clearly see from this definition that since external torque on the body is zero, the angular momentum is going to remain constant. But the angular velocity is not, and that is what which changes with change in angular momentum, because: $$\vec{L\,} = I\vec{\omega\,}$$ For example, ice skaters when have their arms outstretched, their moment of inertia is high and so angular velocity is low, but if they draw in their arms, their moment of inertia decreases and correspondingly, without any external torque, their angular speed increases!

###Edit: The revision to your question has made it further interesting. Imagine that the rod is connected to a motor. Now, once the insect starts crawling towards the end, the moment of inertia of the entire system increases. According to our equations, the angular speed of the system should correspondingly decrease. But, we are told that it remains a constant $\omega$. This means that the motor has continuously apply a torque to keep the angular velocity constant! This is the external torque that we have find in the question, and it is responsible for the increasing angular momentum as well as kinetic energy of the system.

In the first case, there was no constraint keeping the angular velocity constant, unlike your second question.