What Taylor approximation is used in this equation? In Wickens' Macroeconomic Theory book, in page 48(1st edition), the author states that by doing a taylor approximation we get the following result.

Why is that? What approximation did he use?
I've tried several to linearize the fraction $\frac{(1+\eta)^{\sigma}}{\beta}$ around those values, but I don't get the same...
Even though your question does not allow a definite answer, I am pretty sure the author used a Taylor expansion around the logarithm of both sides of the equation. This process is called log-linearization, and is fairly common.

We can approximate (logs of) growth rates as $x \approx log(1+x)$ when $x$ is small. (and by the rules of logarithm:  $\log((1+x)^\sigma ) \approx\sigma x$

Rewriting your equation 
$1+(r_t-\delta)=(1+\eta)^{\sigma}(1+\theta)$

where
$r_t=\alpha (k^{\# *})^{\alpha -1}$

and applying this rule gives

$r_t-\delta=\theta+\sigma \eta$

which gives the desired result.