The tanh function involves two functions, $\sinh{x}$ divided by $\cosh{x}$. This animation illustrates this:

When $x$ is zero, $\sinh x$ is 0 and $\cosh x$ is 1, so the result is 0.

As $x$ gets larger, as the animation shows, $\sinh x$ and $\cosh x$ become ever closer is value, although $\sinh x$ is always slightly smaller. The value of $\tanh x$ therefore tends towards 1.

When $x$ is large and negative, $\sinh x$ becomes ever closer to $-\cosh x$. The value of $\tanh x$ therefore tends towards -1.