It all comes down to a redirect
A link shortener stores a short code that maps to a longer destination URL. When you open the short link, the shortener's server looks up that code and answers with a redirect to the real address.
In the simplest case this is instant. The complexity comes from everything the shortener layers on top of that single redirect to earn money from your visit.
Why the wait and the ads exist
Many shorteners pay their owners per visit, so they insert a countdown, a 'get link' button and one or more ad pages before they finally redirect you. The longer you stay and the more ads you load, the more they earn.
That is why opening a single short link can mean clicking through several pages, dismissing pop-ups and waiting out a timer before the real destination appears.
How resolving skips the busywork
A resolver follows the same path a browser would, but in the background. It waits out the timer, clears the intermediate steps and reads the final redirect target, then hands you that real address directly.
The destination is identical to what you would reach by clicking through manually; you just skip the ads and the wait. Paste a link on the supported sites page to see it happen.