Let $f$ and $\alpha$ be two real functions of real parameters in $[a,b]$. Furthermore, let $P^*=(P,\xi)$ be a dotted partition, $P=\{a=t_0<t_1<\dotsb <t_n=b\}, t_{i-1}<\xi_i<t_i$. We denote the Stieltjes sum as $\sum_{i=1}^n f(\xi_i)[\alpha(t_i)-\alpha(t_{i-1})]$ and define the Riemann–Stieltjes integral as $$\lim_{|P|\rightarrow 0}\sum_{i=1}^n f(\xi_i)[\alpha(t_i)-\alpha(t_{i-1})]=\int_a^b f(t)d\alpha(t).$$ This is essentially the Riemann integral but generalized to a "partition difference" that is given by the difference of function at consecutive points of the partition ($\alpha$ in this case) and as such reduces to the Riemann integral when $\alpha(t)=t$.
My question is: how should I conceptualize this integral? The Riemann integral can be conceptualized as a kind of oriented area/volume for real functions; the Stieltjes integral doesn't seem to have that kind of analogy. What is going on here? Are we integrating $f$ with respect to $\alpha$ (whatever that means)? And would it be correct to say we are integrating $f$ "in the direction of $\alpha$", in some kind of "function space"?
P.S- I have read the selected response in this post and it was most enlightening but I still would appreciate some other kind of approach.