Where can I start learning about higher dimensions in mathematics? As the title suggests, I am looking to gain a good understanding of what higher dimensions are and how they operate, particularly in mathematics. And how I can build an intuitive understanding of this concept. I'm a college student and top-performing at mathematics for those wanting to know my skills/qualifications level.
It would also be really helpful if someone could explain what a "higher dimension" even constitutes as.

The question is too large to answer in generality, but here are a few examples to show the surprising power of modeling $n$-dimensional Cartesian (linear combinations only, the first two items below) or Euclidean space (including the standard dot product, permitting definitions of length and angle, the third example) by ordered $n$-tuples of real numbers.

*

*Two planes in four-space can meet in a single point. Consider Cartesian $4$-space with coordinates $(u, v, x, y)$, The sets
$$
P_{1} = \{(u, v, 0, 0) : \text{$u$, $v$ real}\},\qquad
P_{2} = \{(0, 0, x, y) : \text{$x$, $y$ real}\}
$$
are planes, intuitively because each is a copy of plane Cartesian coordinates with a couple of $0$s tacked on. These planes meet at a single point, $(0, 0, 0, 0)$. This cannot happen in three-space, where any two non-parallel planes meet in a line.


*A plane in four-space does not separate, any more than a line separates three-space. Using the notation of the first item, the set
$$
C_{1} = (\cos t, \sin t, 0, 0) : \text{$t$ real}\} \subset P_{1}
$$
is a circle in the plane $P_{1}$, and "links" the plane $P_{2}$. Particularly, we can travel from $(0, 0, 1, 0)$ to $(0, 0, -1, 0)$ without passing through $P_{2}$.


*A (hyper-)cube with $1$cm sides in a sufficiently high-dimensional Euclidean space can hold a washing machine, or the Eiffel tower, or the Oort cloud. For definiteness let's think of a washing machine as fitting inside a three-dimensional cube whose sides are $100$cm in length. The crucial ingredient is the Pythagorean theorem in Euclidean $n$-space, which guarantees that the distance between two ordered $n$-tuples of reals is the magnitude of their vector difference, i.e., the square root of the sum of the squares of the differences of their coordinates. For instance, if
$$
p_{1} = (1, 1, 0, 0),\qquad
p_{2} = (-1, 1, 2, -1)
$$
are points of Euclidean $4$-space, then
$$
p_{2} - p_{1} = (-1, 1, 2, -1) - (1, 1, 0, 0)
= (-2, 0, 2, -1)
$$
has magnitude $\|p_{2} - p_{1}\| = \sqrt{(-2)^{2} + 0^{2} + 2^{2} + 1^{2}} = \sqrt{4 + 0 + 4 + 1} = \sqrt{9} = 3$. Similarly, the magnitude of $p_{1}$ itself, i.e., the distance from the origin to $p_{1}$, is $\sqrt{2}$ and the magnitude of $p_{2}$ is $\sqrt{7}$. The origin and the points $p_{1}$ and $p_{2}$ are vertices of a right triangle with these sides; we can calculate the interior angles using trigonometry, while getting a protractor into $4$-space is ... inconvenient. (There are easier ways to calculate angles in $n$-space using the Euclidean dot product, which can be found in numerous answers on-site.)
Now let's think about the vector $(1, 1, \dots, 1)$ with $n$ components all equal to $1$. Its magnitude is $\sqrt{n}$, which can be made as large as we like. If our unit is $1$cm, then taking $n = 10,000$ (so $\sqrt{n} = 100$) gives a vector one meter long whose components are individually all $1$cm. Maybe you can see where this is heading.
To fit in our entire washing machine, it suffices to work in $30,000$-dimensional space: Writing $\mathbf{0}$ to denote a list of $10,000$ zeros and $\mathbf{1}$ to denote a list of $10,000$ ones, consider the three vectors
$$
  p_{1} = (\mathbf{1}, \mathbf{0}, \mathbf{0}),\qquad
  p_{2} = (\mathbf{0}, \mathbf{1}, \mathbf{0}),\qquad
  p_{3} = (\mathbf{0}, \mathbf{0}, \mathbf{1}).
  $$
Each vector has length $100$(cm), and any two are perpendicular (the dot products are pairwise $0$). The cube they span, i.e., the $3$-dimensional set of vectors
$$
  xp_{1} + yp_{2} + zp_{3} = (x\mathbf{1}, y\mathbf{1}, z\mathbf{1})
  $$
in $30,000$-dimensional Euclidean space, contains three mutually-perpendicular one-meter segments, so it can hold a one-meter cube. Finding $1$cm cubes to hold larger objects is left as a pleasant exercise.
High-dimensional geometry is mind-bending at first, but becomes mind-expanding, and eventually as familiar in some ways as spatial geometry of the external world.
