Why do Fourier Series work? I would like to have an intuitive understanding of Fourier Series. I mean, I know the formulas:
$$
f(t) =\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(n\pi tL)+\sum_{n=1}^\infty b_n \sin(n\pi tL)
$$
And given a Wave, I know how to do the calculations, however, I have no idea why this formula works besides the fact it sums an infinity of sines and cosines.

How did Fourier arrive at this formula? And what's it meaning? I would be thankful if one could give me a brief explanation or point me reference material.

Thanks in advance
$\newcommand{\Vector}[1]{\mathbf{#1}}\newcommand{\vece}{\Vector{e}}$The linked questions provide good answers, but may be at the technical end of "intuitive". Here's a fast-and-loose conceptual motivation:

If $\bigl(V, \langle\ ,\ \rangle\bigr)$ is an $N$-dimensional real inner product space, and if $\{\vece_{n}\}_{n=1}^{N}$ is an (ordered) orthonormal basis, then an arbitrary vector $v$ in $V$ may be written as a linear combination
$$
v = \sum_{n=1}^{N} \langle v, \vece_{n}\rangle \vece_{n}.
\tag{1}
$$
Indeed, $\{\vece_{n}\}$ is a basis of $V$, so there exist real coefficients $a_{k}$ such that
$$
v = \sum_{k=1}^{N} a_{k} \vece_{k}.
\tag{2}
$$
Taking the inner product of each side with $\vece_{n}$ gives
$\langle v, \vece_{n}\rangle = a_{n}$ because the basis $\{\vece_{n}\}$ is orthonormal.

Loosely, one might expect a similar conclusion to hold if $V$ is infinite-dimensional. Getting the definitions and hypotheses right, and proving a version of (1) in this new setting, is why any "honest" answer is bound to be technical. Phrases in quotes below are not mathematically correct, and therefore require careful inspection and/or justification.

Intuitively, let $L &gt; 0$ be real, let $V$ be "the space of real-valued functions" on $[-L, L]$, and define an "inner product" by
$$
\langle f, g\rangle = \frac{1}{L} \int_{-L}^{L} f(t) g(t)\, dt.
$$
The functions
$$
C_{n}(t) = \begin{cases}
1/\sqrt{2}, &amp; n = 0, \\
\cos(n\pi t/L), &amp; n &gt; 0;
\end{cases}\qquad
S_{n}(t) = \sin(n\pi t/L),\quad n &gt; 0;
$$
turn out (by elementary calculus and trigonometry) to form an "orthonormal basis" of $V$.

Loosely, we expect that if $f$ is a function, we can express $f$ as an infinite sum of these basis functions, and the coefficients are the inner products of $f$ with the basis elements, i.e. (for $n &gt; 0$),
\begin{align*}
a_{0} &amp;= \langle f, 1\rangle = \frac{1}{L} \int_{-L}^{L} f(t)\, dt, \\
a_{n} &amp;= \langle f, C_{n}\rangle = \frac{1}{L} \int_{-L}^{L} f(t) \cos(n\pi t/L)\, dt, \\
b_{n} &amp;= \langle f, S_{n}\rangle = \frac{1}{L} \int_{-L}^{L} f(t) \sin(n\pi t/L)\, dt, \\
f(t) &amp;= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} (a_{n} C_{n}(t) + b_{n} S_{n}(t), \\
  &amp;= \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n} \cos(n\pi t/L) + b_{n} \sin(n\pi t/L).
\end{align*}
(The "special" factor of $1/2$ on the constant term arises because $C_{0} = 1/\sqrt{2} \neq 1$.)
Fourier was looking at guitar strings. He reasoned that the waveform (of the string) changes with time and the amplitude at the ends are allways zero. He intuitively reasoned that any waveform of the strings can be represented as a weighted sum of sine/cosine waves that can fit into the guitar by having zero amplitude at the ends (i.e. the harmonics). He then proposed that by removing the restriction that the amplitude must be zero at the ends, any continuous function can be represented as a sum of sine/cosine waves (waves with different wavelengths/frequencies).