Le quotidien (noun) has a different meaning than the adjective, even of course if the meaning is related.

In French you can turn almost any adjective into a noun by simply putting an article in front of it. And the meaning of the resulting noun is always related (not identical) to that of the adjective.
Let’s take the adjective grand. You can talk of les grands as for instance :

• C’est plus facile pour les grands d’attraper les boites en haut des rayons de supermarchés.
  Les grands refers to all tall people.

Quotidien as a noun can be two different things but it is clear in your sentence it refers to la vie quotidienne. All French people understand it without the need to add vie.¹

Note that quotidien as a noun can have two different meanings. In a different context it could mean un journal quotidien (a newspaper that is published every day):

• Je sors acheter mon quotidien.

is also a sentence every French person would understand without having to precise they are talking about a newspaper.

¹ But I disagree with you about the English: “you forget about the daily”, it would not mean “you forget about daily things/life”. In English, a “daily” with no further detail is either a (daily) newspaper, or a (daily) help who does your housework.

What exactly does “le quotiden” mean in “tu oublies le quotiden”?

That means “You forget about the nitty-gritty of everyday life.”