The linearity here means that your model is a linear combination of the predictors, e.g. in autoregressive model $AP(p)$: $$ X_t = c + \sum_{i=1}^{p}\phi_i X_{t-i} + \epsilon_t $$ Where $\phi_1,...,\phi_p$ are your model parameters, $\epsilon$ represents noise and $c$ is some constant. That equation says that if you want to predict the value of $X_t$ you simply take a linear combination of some $p$ past $X$ values (weighted by your model parameters $\phi$). Given an initial (time series) set of $\{X_{t-p}, ... X_{t-1}\}$ there is no reason to assume that plotting $\{X_{t}, ...\}$ will give you a straight line.

The linearity here means that your model is a linear combination of the predictors, e.g. in autoregressive model $AP(p)$: $$ X_t = c + \sum_{i=1}^{p}\phi_i X_{t-i} + \epsilon_t $$ Where $\phi_1,...,\phi_p$ are your model parameters, $\epsilon$ represents noise and $c$ is some constant. That equation says that if you want to predict the value of $X_t$ you simply take a linear combination of some $p$ past $X$ values (weighted by your model parameters $\phi$). Given an initial (time series) set of $\{X_{t-p}, ... X_{t-1}\}$ there is no reason to assume that plotting $X_{t}, ...$ will give you a straight line.