From $[K:\mathbb{Q}]=4 , $ I know that there are are $4$ embeddings {$\sigma_1 ,\sigma_2,\sigma_3,\sigma_4$} .
Furthermore, each element in $K$ can be written as $$ a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \ \ a,b,c,d \in \mathbb{Q}$$
So first I have the identity $$\sigma_1(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6})=a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} $$
Then I have to consider $$ \sigma_2(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6})$$
I know that I have to take the conjugate of some terms , here for $\sqrt{2}$ and $\sqrt{3} $ , but I do not really understand why I have to consider these roots .
Well the extension is generated by $\sqrt 2$ and $\sqrt 3$, so the embedding is defined (generated) by the images of these two numbers.
The image of $\sqrt 2$ has to satisfy $x^2-2=0$ and the image of $\sqrt 3$ has to satisfy $x^2-3=0$ (these being the minimal polynomials of the two roots over $\mathbb Q$).
These basic facts define the options for embeddings, which you should be able to identify.