I kept seeing articles about the drawbacks of R-squared (and that's why we need to have adjusted R-squared).

One drawback is that: "Every time you add a predictor to a model, the R-squared increases, even if due to chance alone. It never decreases. Consequently, a model with more terms may appear to have a better fit simply because it has more terms." (link).

Mathematically, why does this happen? And mathematically, why adjusted R-squared can solve this problem?