By definition, the strings in $X^{*}$ (for any language X, whether X=∅ or not) are those constructed by taking some finite number (possibly 0) of strings from X and concatenating them.
If you take 0 strings and concatenate them, you get ε. Note that this has nothing to do with whether X=∅ or not. The empty string ϵ is always in $X^{*}$ regardless of what X is.
When X=∅ there are no other strings in $X^{*}$, because you cannot take more than 0 strings from ∅. So the only string in $\varnothing ^{*}$ is ϵ: thus $\varnothing ^{*}$ ={ϵ}.
Now let X be some nonempty language, say X={a}. Then $X^{*}$={ϵ,a,aa,aaa,aaaa,…}. Notice that ϵ is still in $X^{*}$, but now there are other strings because I can concatenate one or more a's together. The ϵ is what I get by concatenating zero a's.
