You’re right that this is counter-intuitive. However, I find the following analogy insightful:
Suppose you have a parallelepiped that is $2\times 3\times 1$, with all the side lengths measured in inches. What are the side lengths, as measured in centimeters?
To answer this question, we have to calculate how many centimeters comprise $2,3,$ and $1$ inches respectively, just like in the case of changes of basis. The reason for this is that the object you’re measuring is staying the same, but the language with which you’re describing it is changes. So you need to work backwards, and ask yourself “how do I say the same thing I just said, but in my new language.”
This is very different from the case of if the object was to change size, and then need to be expressed in the original language. You might express that as a problem as follows:
Suppose you have a uniform density parallelepiped of ice that is $2\times 3\times 1$, with all the side lengths measured in inches. After being left out in the sun, it melts so that it loses half of its total mass, and shrinks equally in every dimension. What are the new dimensions of the cube?
Here the universe didn’t change, just the object in question, and so you have to find the size of the object in terms of the old basis. Of course, we could combine these two types of problems and make you do both simultaneously.
In my answer to your other question, I pointed out that your textbook didn’t name the linear function, only it’s matrix representation with respect to the standard basis. The key insight here is that the metric representation isn’t the base thing that exists in the world, rather it’s a description of the thing that exists in a particular language, just like $2in\times 3in\times 1in$ is. The true underlying object, the parallelepiped, is the linear transformation. The units we measure it in, the language we use to talk about the properties of the thing, is the matrix representation.