For any $\theta \in \Theta$, write $\ell(\theta) - \ell(\theta_0) = \mathbb{E}_{\theta_0}\!\left[\log \frac{f(X, \theta)}{f(X, \theta_0)}\right]$. Apply Jensen's inequality using the concavity of $\log$: since $\log$ is concave, $\mathbb{E}[\log Z] \leq \log \mathbb{E}[Z]$ for any non-negative random variable $Z$. Therefore \begin{align*} \ell(\theta) - \ell(\theta_0) \leq \log \mathbb{E}_{\theta_0}\!\left[\frac{f(X, \theta)}{f(X, \theta_0)}\right] = \log \int_{\mathcal{X}} \frac{f(x, \theta)}{f(x, \theta_0)}\, f(x, \theta_0)\, dx = \log \int_{\mathcal{X}} f(x, \theta)\, dx = \log 1 = 0. \end{align*} The final integral equals $1$ because $f(\,\cdot\,, \theta)$ is a probability density for every $\theta \in \Theta$. For the discrete case replace the integral by a sum; the argument is identical.