let bar: Result<T, E> = ...;
let foo = bar.inspect(|value| log::debug("{}", value));

is equivalent to

let bar: Result<T, E> = ...;
let foo = bar.map(|value| {
    log::debug("{}", value);
    value
});

Looks vaguely like Stream::peek from Java, I think? There's an equivalent method in Iterator::inspect.

it's just a way to use map with a reference instead of the value, by what I understood.

could be usefull for logging values in a Result so you can see it. However I think you can already do that by just mapping and returning the variable.