Darts in Higher Dimensions (with 3blue1brown) - Numberphile

You can and as the dimensions go up the volume of the inscribed ball as a percentage of the volume of the hypercube shrinks down a lot. There's a nice video about this from Numberphile and 3blue1brown - ▶ this video

Higher dimensional geometry can show up in lower dimensional problems. This numberphile video ( ▶ this video ) involves a puzzle about throwing darts at a dart board which is solved by using the volumes of 4+ dimensional spheres.

In a unique Numberphile video featuring Grant Sanderson (3blue1brown), this weird number pops up in a game of darts. [0] ▶ this video

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 - ▶ this video Or 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.