The Sanskrit words which are used in Advaita doctrines to describe what you called "illusion" are MithyA, MAya, AvidyA etc. We need to see the definitions of these words within Advaita. They have terminological usages within the framework of Advaita Vedanta. The usual literal meanings are not applicable.
Definition of mithyA:
BhAtiti ched bhAtu nAma bhushanam mAyikaya tat |
Yad-sad-bhAsamAnam tan-mithyA swapna-gajAdi-vat ||
That which does not exist in truth but appears to exist - that is
called mAyA or mithyA like the entities like elephants etc which are
viewed in dreams.
So, within Advaita Vedanta mithyA= something that does not exist in truth but appears to exist. Thus, there is no contradiction if we are being able to see this world. Even Advaita says that it will appear until the point when one obtains Samyak GyAna (Knowledge of the Self).
Definition of mAyA:
Ritehartham yat pratiyate na pratiyate chAtmAni |
mAyAm ythAbhAso yathA tamah ||
Like a reflection or like RAhu (mentioned as Tamah in the verse) that
which appears to be existent without any subject but which does not
appear on the Atma is MayA.
Bhagavata PurAna 2.9.33
Another definition quoted in YogavAshishtasArah is:
YA mA kintu pratiyate sA mAyA - That which actually does not exist but
appears to exist is mAyA.
So, when they say "Jagat mithyA" it does not imply that this world will not be visible to us or something like that. Because MithyA has a special meaning within the Advaita doctrine.