The maximum over a restricted set is mathematically no larger than the maximum over the full set. You can view the maximized likelihood for model with fewer regressors as the maximum over a restricted set.
Specifically, if you have three regressors, the parameters are $(\beta_0, \beta_1, \beta_2, \beta_3)$, and the maximized likelihood is the maximum over all possible combinations of $(\beta_0, \beta_1, \beta_2, \beta_3)$. The restricted model having only one regressor (say $X_1$) has maximized likelihood over the same set of combinations $(\beta_0, \beta_1, \beta_2, \beta_3)$, but restricted so that $\beta_2 = \beta_3 = 0$. The maximum over the restricted set is no larger than the maximum over the unrestricted set; in most cases it is smaller.
Just because the maximized likelihood is smaller does not necessarily mean the model is worse, though. Since this occurrence is a mathematical fact, the unrestricted model will have (ordinary) higher maximized likelihood even when $\beta_2 = \beta_3 =0$ in reality. The likelihood ratio test specifically addresses this issue, providing a reasonable answer to the question as to whether the difference in maximized likelihoods is explainable by chance alone.
Even if $\beta_2 \neq 0$ or $\beta_3 \neq 0$, the model with only $X_1$ still might be better; penalized likelihood and out-of-sample predictions address this issue.