Don't stay up late just because you're not sleepy.
Literal
Not-sleepy [because-から] [quotative-と] say [only-だけ] [for-で] late-until be-awake prohibitive-[てはいけない].
~からと言うだけで = 'just because (someone) says X.' Combines から (because) + と言う (to say) + だけで (only with/by) for 'just on the grounds of saying X.' 遅くまで起きている = 'stay up until late.' An admonition against making excuses.