Putting together a layman's answer/example.
To further simplify, I will use the example of a coin flip to emphasize the results.
Given a single coin, you have a 50% change of flipping a "heads".
Given a second coin (or flipping the same coin a second time), you now have a separate and independent 50% chance for each coin flip.
With two coin flips you now have four possible results, as shown in this graphic:
Each result has a 25% change of occurring, but 3 of the results include at least a single "heads" result. To get the likelihood of at least 1 "heads" you simply add together the likelihood of each result that includes a "heads". In this case, your chance of rolling at least 1 "heads" is 75%.
If you add a third coin (or flip the same coin a third time), you now have eight possible results, as shown on this graphic:
Each individual results has only a 12.5% chance of occurring, but 7 of the 8 results include at least a single "heads" result. Adding these together you get an 87.5% chance of getting a single heads.
Each additional coin-flip results in a greater and greater chance of getting at least a single heads.
Returning to using a 10-sided die, the results follow the same trend, though with a 10% chance of any single roll giving your wanted result, the increase is much less pronounced, though still significant.
A single die gives you your 10% chance.
Two dice (or a single die rolled twice) gives you 100 possibilities, of which 19 have at least a single 1 (19% chance).
Three dice (or 3 rolls) gives you 1000 possibilities, of which 271 have at least a single 1 (27.1% chance).
Source of images is math-prof.com