Higher dimensions can be imagined by looking at how you view lower dimensions. For e.g.: If you want to imagine something 4D, you need to understood how we interact with 2D objects. How would we appear to an entity who is 2D. They can only see edges (actually this edge too had 0 thickness, but for practical purposes lets assume they can see very thin edges like the thickness of a paper). Such 2D entities can never see us in our 3D entirety.
For e.g. say a circular 2D entity and we pass our finger through a paper, they would just see a circle forming with increasing diameter till its circumference is equal to that of our finger and then when we take our hand off, this circle would reduce in size and then vanish.
You(a 3D entity) can touch the center of the circle(2D entity) without breaking its edges. In a similar way, maybe a 4D entity could touch our inner organs without cutting us open.
The movie “Flatland” portrays this concept really well of 2D entities interacting with a 3D entity.
This can be supplemented with further reading about the allegory of “Plato’s cave”.