i've seen a few people leave more algebraic/technical explanations so i thought i would try to give a more handwavy explanation. there are three things we need:

1. the sum of two numbers doesn't depend on how those numbers are written. (for example, 1/2 + 1/2 = 0.5 + 0.5.)
2. 1/3 = 0.33...
3. 1/3 + 1/3 + 1/3 = 1.

combining these three things, we get 0.99... = 0.33... + 0.33... + 0.33... = 1/3 + 1/3 + 1/3 = 1.

it's worth mentioning the above argument could be refined into an actual proof, but it would require messing around with a formal construction of the real numbers. so it does actually explain "why" 0.99... = 1.