A mathematician's guided tour through higher dimensions
I think it's easier to approach higher dimensions without talking about spatial dimensions at all.For example - Darts in Higher Dimensions, 3blue1brown and Numberphile - https://www.youtube.com/watch?v=6_yU9eJ0NxAOr even more trivially, you can think of a table where every row is some entity and every column is some attribute associated to it. For example, make a spreadsheet where each row is a person and the columns are age, height, weight, salary, and years to retirement - then you can think of each person as a point in 5-d space. And some properties are intuitively obvious - for example as you keep adding more columns it becomes more difficult to find people who are similar to each other. It's a pretty accessible way to introduce high dimensions without talking about tessaracts.