Answer:
Clyde discovered Pluto with a device called a "blink comparator." In its time, it was top-of-the-line technology, circa 1930.
Full-sky or All-sky surveys were taken from time to time on old-fashioned photographic plates back in the day. Most stars move imperceptibly over several human lifetimes, because they are so far away; but planets, they are much closer to Earth. Even distant Pluto moved enough to be captured by 1930's technology.
The blink comparator focused the image of two different photo plates taken at different times, but of the same region of the sky, rapidly back and forth. Any planet, or "wandering star" from the Greek language, would manifest itself as movement in the rapidly changing views of the comparator. Easy enough for a human to detect.
Especially one so talented as Clyde Tombaugh. And that is how he found it.