The word "just" has many uses. In this example "just' is used to emphasize that you only need to sleep or that you need nothing more than to sleep. Meanings of "just": 1 : to an exact degree or in an exact manner 2 a : very recently b : at this or that exact moment or time c — used to say that two or more events are happening at the same time or with very little time between them d — used to emphasize that a moment or time is not far from the present moment 3 a : by a small amount b : by a small distance 4 a : nothing more than : only b — used to stress the simple truth of a description or statement c — used in polite requests d — used for emphasis when you give an order or make a suggestion e — used to describe what someone does instead of doing what is necessary or expected 5 — used with words like might and may to refer to something that is possible For examples of this definition: http://www.learnersdictionary.com/search/just%5B2%5D.

"I need to sleep" - suggests that you need sleep but you also might need to finish your work first, or get home, or any NUMBER of things but sleep is not the only important thing. "I just need to sleep" means that either there is only one thing you need "sleep" or amongst all of the the things the most important is sleep. "I can't stop my eyes from closing. I just need to sleep."