If the compact simplex is
$$\Delta_n = \{ (x_0,\cdots,x_n) : x_i \geq 0, x_0+x_1+\cdots+x_n=1\}$$
then consider this function $f : \Delta \to \mathbb R \cup \{\infty\}$ defined by
$$f(x_0,\cdots,x_n) = \frac{1}{x_0} + \cdots + \frac{1}{x_n}$$
This is a proper Morse function on the interior of $\Delta_n$, and there's only the one critical point at $(\frac{1}{n+1},\cdots,\frac{1}{n+1})$, so standard theorems in Morse theory give you a diffeo to the open ball.
I imagine this is simple enough that you could solve the corresponding ODEs explicitly and write the diffeo out in a closed-form but I haven't put in the work.