Let $\gamma:I\rightarrow\mathbb{R}^n$ be a parameterisation of some regular curve $C$. The definition I am working with states that a parameterisation $\tilde{\gamma}:J\rightarrow\mathbb{R}^n$ of a curve $C$ is a reparameterisation of $\gamma$ if there is a diffeomorphism $\phi:J\rightarrow{}I$ such that $\tilde{\gamma}(\tilde{t})=\gamma{}\space\circ\space\phi(\tilde{t})$ for all $\tilde{t}\in{J}$.
Now in my lecturers notes, he introduces arc length parameterisation in the following:
Is there an error? Isn't $s$ the diffeomorphism, not the reparameterisation? I'm confused as to how he has obtained the last line, and which is the original parameterisation, which is the reparameterisation and which is the diffeomorphism.
If $\gamma: I \to \mathbb{R}^n$ is the (original) parametrisation of a curve, then
$$s: I \to J \subset \mathbb{R}: t \mapsto s(t) = \int_{t_0}^t \|\gamma_t(t)\|\,dt$$
is the arc length of the curve $\gamma$ from $t_0$ to $t$; it is really just the length of the piece of curve between the points $\gamma(t_0)$ and $\gamma(t)$.
Writing $\gamma(t) = (x_1(t), \ldots, x_n(t))$, its derivative $\gamma_t(t) = (\dot x_1(t), \ldots, \dot x_n(t))$ is the velocity and the norm of this is the speed $\|\gamma_t(t)\| = \sqrt{(\dot x_1(t))^2 + \cdots + (\dot x_n(t))^2}$. Note that $\frac{ds}{dt} = |\gamma_t(t)|$.
In this setting the diffeomorphism $\phi$ is not $s(t)$, but it is the inverse of the $s(t)$, namely $\phi(s) = t(s)$ (the parameter $s$ plays the role of $\tilde t$ in your definition). Note that this is indeed a function $J\to I$. So $\tilde \gamma = \gamma \circ t(s)$ is the reparametrisation.
The last line follows from the chain rule: $$ \tilde\gamma_s(s) = \frac{d}{ds} (\gamma \circ t(s)) = \gamma_t(t(s)) \circ \frac{dt}{ds}. $$ So the speed of the reparametrisation is $\|\tilde \gamma_s(s)\| = \|\gamma_t(t(s))\| \|\frac{dt}{ds}\| = \|\frac{ds}{dt}\| \|\frac{dt}{ds}\| = 1$.
The intuition behind this is very natural: suppose you are traversing a curve with a non-constant speed; that is the original parametrisation $\gamma(t)$. The point $\gamma(t)$ is the point on the curve you reach after traveling a time $t$ seconds. The re-parametrisation $\tilde \gamma$ is like traversing the curve with a constant speed $1\, \mathrm{m/s}$. The point $\tilde \gamma(s)$ is the point that lies at a distance $s$ meters from the starting point.