The most I can visualize or perceive are 4 dimensions. Yes, 4, because I can also watch videos (which have 3 spatial dimensions and 1 temporal one). Remember Einstein's spacetime?
When dealing with $n$-dimensional spaces, for $n > 4$, I simply do not care about visualizing them in my head, but, as someone suggests, we can think of them as "degrees of freedom". Maybe something like a tesseract may be interesting to you, but that's not really useful to me, to be honest.
When dealing with the math that involves $n$-dimensional spaces or objects, you often do not have to visualize anything, but just have to apply the rules. For example, if you are multiplying multi-dimensional arrays, you just need to make sure that the external dimensions match, and stuff like that.
There are cases, when dealing e.g. with TensorFlow's tensors, where you can imagine that there are matrices for each of the elements at that coordinate of the tensor, but that's not very common.
In case you really want to visualize $n$-dimensional objects, you could first project them to $2$ or $3$ dimensions with some dimensionality reduction technique (e.g. t-SNE).