I know 也 only as "also", but I see it now and then used in ways difficult to understand, especially in some negative statements. Why is this so? Does 也 also have some meaning not mentioned in dictionaries which is used in these cases?
什麼都 can be used in both positive and negative statements while 什麼也 is usually used in negative statements. So, instead of focusing on the usage of 也 here, 什麼都 and 什麼也 can be regarded as phrases which are sometimes used interchangeably.
One possible translation:
We thought we knew/know everything but, in fact, we knew/know nothing at all.
So, it's ok to say (although the above example would read better):
but odd to say:
The meaning of the original expression is "I have nothing." But Chinese gets there in a "strange," way.
A more literal translation of the expression is "Everything not have." The "everything" is rendered by "什麼都," which literally translates to "whatever all."
什麼也 gets to the same place in a slightly different way. I would translate it as "whatever included." And a literal translation would be "whatever included, not have," instead of "whatever all" not have.