Oh sure, I bet karpathy's explanation is great for the people it makes sense to. It just assumes a lot more specialized knowledge on the part of the reader, knowledge that I personally don't have.
Consider the space around you. Pick some frame of reference with a fixed origin point in space. To specify an object, you must specify the distance along each independent direction (x, y, and z in 3d space) to arrive at the object from the fixed origin. These independent directions are referred to as a basis for the space. Often these independent bases are perpendicular (or orthogonal).
For higher dimensional spaces its the same idea. It turns out we can even have infinite dimensional spaces whose bases are functions that can be perpendicular in a certain sense. It also turns out that complex exponentials (waves) are an orthogonal basis for a certain space of functions. This is really cool! It means we can refer to any function in this space using the complex exponential basis functions just like we referred to objects in 3d space using three independent directions.
Why is this useful? The magnitude in a particular independent direction in this space tells us about frequency content of that function. So instead of the first coordinate telling you how much x distance there is, it says how much of a certain frequency there is.
The Fourier transform is usually used to flip from the time domain representation to the frequency domain representation.
"Time domain representation" means when you plot that information, the x-axis is in units of time; seconds or fractions of seconds, usually, for something humans can perceive as sound. The line tells us how loud the signal is at a given instant of time.
"Frequency domain representation" means when you plot that information, the x-axis is in units of frequency; hertz or kilohertz, usually, for something humans can perceive as sound. The line tells us how loud the signal is at a given frequency.
Here's the bit about basis vectors: The frequency domain representation is possible because we can make any graph by adding enough sine functions, assuming we're allowed to do things like increase or decrease their magnitude, increase or decrease their frequency, and shift them left or right relative to each other (adjust their phase).
Therefore, we can describe the graph were interested in using only sine waves, which are pure frequencies, and when we've done that, we'll know which specific frequencies contributed most to the information we represented in the original graph.
There is, of course, an inverse Fourier transform, which takes a frequency domain representation to the time domain, so we can do things like record a sound (time domain), take it into the frequency domain, remove certain frequencies (typically all frequencies above and/or below some cutoff) by zeroing parts of the graph, and take the result back into the time domain, so it can be played back as a recording.