This can be most easily seen through Lagrange duality: there exists some $C$ so that $$\arg\min_{\beta \in \mathbb{R}^p} RSS + \lambda \sum_{i=0}^p \beta_i^2 = \arg\min_{\beta\in\mathbb{R}^p \, : \, \|\beta\|_2 \leq C} RSS.$$ Further, we know that larger $\lambda$ corresponds to smaller $C$. Therefore, increasing the tuning parameter $\lambda$ further constrains the $\ell_2$ norm of the coefficients, leading to less flexibility.

This can be most easily seen through Lagrange duality: there exists some $C$ so that $$\arg\min_{\beta \in \mathbb{R}^p} RSS + \lambda \sum_{i=0}^p \beta_i^2 = \arg\min_{\beta\in\mathbb{R}^p \, : \, \|\beta\|_2^2 \leq C} RSS.$$ Further, we know that larger $\lambda$ corresponds to smaller $C$. Therefore, increasing the tuning parameter $\lambda$ further constrains the $\ell_2$ norm of the coefficients, leading to less flexibility.