This sentence “Brahman is real, this world is unreal” is the core teaching of Advaita Vedanta philosophy. Upanishads has mainly highlighted this teaching only.
In Vivekchudamani verse 20, Adi shankaracharya has said – ब्रह्म सत्यं जगन्मिथ्येत्येवंरुपो – “Brahman is real , this world is an illusion”
Brahman is Real – It means that Brahman ( the supreme creator) and Jivatma (Individual) are one entity.
Brahman is acting as Jivatma under the influence of Maya and consider him as a limited body, who is subject to birth and death.
Brahman is formless, space-less, timeless, changeless, ever present and is the source of this world and its objects.
World is Unreal – This world is creation of Maya and is false. This world is a manifestation of Brahman due to his own power called Maya. Similar to the dream world projected by the mind during sleep, where an individual takes himself and dream world both as real.
Maya is the power which block correct perception of an object and makes it look different. In case of mirage in desert, one see water but there is no water, only sand is seen as water. Similarly this Brahman (sand) is seen as world (water)
Vedanta also have another concepts which says that Brahman is non-dual, there is nothing separate from him. Even Maya is false and it has no existence apart from the Brahman.