There are various ways to make this rigourous. The one that requires the less machinery is to simply use the fundamental theorem of calculus as it is done here. In this approach, the $dt$ is purely notation for the integration variable. The proof goes
Let $f$ and $\phi$ be two functions satisfying the above hypothesis that $f$ is continuous on $I$ and $\phi'$ is integrable on the closed interval $[a,b]$. Then the function $f(\phi(x))\phi'(x)$ is also integrable on $[a,b]$. Hence the integrals
$\displaystyle\int _{\phi (a)}^{\phi (b)}f(u)\,du\hspace{10mm}$ and $\hspace{10mm}\displaystyle\int _{a}^{b}f(\phi(x))\phi '(x)\,dx$
in fact exist, and it remains to show that they are equal.
Since $f$ is continuous, it has an antiderivative $F$. The composite function $F\circ\phi$ is then defined. Since $\phi$ is differentiable, combining the chain rule and the definition of an antiderivative gives
$$(F\circ \phi )'(x)=F'(\phi (x))\phi'(x)=f(\phi (x))\phi'(x)$$
Applying the fundamental theorem of calculus twice gives
$$\begin{aligned}
\int_{a}^{b}f(\phi (x))\phi'(x)\,dx
&=\int _{a}^{b}(F\circ\phi)'(x)\,dx\\
&=(F\circ\phi )(b)-(F\circ\phi)(a)\\
&=F(\phi(b))-F(\phi (a))\\
&=\int_{\phi (a)}^{\phi(b)}f(u)\,du
\end{aligned}$$
which is the substitution rule.
In the formalism of differential forms, the substitution can also be thought of as a pullback along a diffeomorphism. Here the integral can be written
$$\int_{[a,b]}w\,\mu$$
where $\mu$ is the volume 1-form on $[a,b]$. I.e. your integral is thought of as the integral of a differential 1-form along a 1-chain. Then you use a diffeomorphism $[\alpha,\beta]\xrightarrow{\phi}[a,b]$ to rewrite this as a pullback
$$\int_{{\phi}([\alpha,\beta])}w\,\mu=\int_{[\alpha,\beta]}\phi^{*}(w\,\mu)=\int_{[\alpha,\beta]}(w\circ\phi)\,\phi^{*}\mu=\int_{[\alpha,\beta]}(w\circ\phi)\,\phi'\,\nu$$
where $\nu$ is the volume form on $[\alpha,\beta]$.
There is also the formalization of the change of variable used in measure theory. That is: you define the pushforward measure from a measure space to a measurable space along a measurable function. You can read about it here.
And so on.
