You might be an intuitive thinker. You need an image, a film, in your head to properly understand and visualise the scenario. I have it the same way. You can visualise a scenario like this by finding a proper example. Eventually with training you will be able to make up an example and see the scenario before you on the spot.
Here, the force grows when the position grows. We might be dealing with a spring - when you elongate it (position change of one end of the spring), the spring force increases linearly according to Hooke's law. Meaning, when you elongate it it gradually becomes harder to elongate it further.
Does anyone have any method or tips on visualizing the movement of a box following this force position chart?
Note that we do not know the movement, the motion, from this graph. You can elongate your spring fast or slow. It doesn't matter. You can elongate it to $3\,\mathrm m$ on the x-axis with a corresponding force on the y-axis, and then keep it there; maybe you tie the spring end to something so it just stays at this setup. In other words, where you are on this graph says nothing about the movement or motion.
When the axes do not involve speeds, accelerations or most importantly time, then you might not have enough info from the chart to determine a unique motion. The graph here just shows us the relationship between force and position of something, but says nothing about any motion parameters.