Note: In the Russell paradox containing does not mean that a set is a subset of itself. The set contains itself as an element (and not as a subset.) Having this is mind let's step forward.
The size of a finite set depends on the number of elements in that set. If the elements are sets then the number of elements in the contained sets do not count when we calculate the number of elements in the containing set. So, let's see the following example:
Here the number of elements of the outer set is $4$ (two sets and two "element-elements") and not $6$, the number of elements we can see from above.
To consider infinite sets the problem is a little more complicated but is essentially similar.
As far as a set containing itself as an element:
Here the number of elements in $A$ is $3$ even if $A$ is contained in $A$ as an element.
The problem is not that with $A$... You can sense the problem if you try to depict $A$ like I did and stopped at the $3^d$ level.