Saying ##f(f(f(x)))=x## is like saying ##f(u\in[0,1])=x##. As such, ##f:[0,1]\to[0,1]## is bijective.
Since ##f## is a continuous bijection on ##[0,1]##, then it is either strictly decreasing or strictly increasing, the same for its inverse (which has the same direction of variation as ##f##).
Proving ##\forall x\in [0,1],\,f(f(f(x)))=x\implies f(x)=x## is the same as proving ##f(x)\neq x\implies\exists x\in [0,1]\,|\,f(f(f(x)))\neq x##.
Suppose that ##f(x)\neq x##, i.e, there exists a real number in ##[0,1]## such that its image by ##f## is different than itself. In the following, by ##a## I mean this real.
All inequalities will be strict because ##f## is a continuous bijection.
Suppose that ##f(x)## is strictly increasing, and let there be ##a\in[0,1]## such that ##f(a)=b\neq a##, then:
If ##b<a##, then ##f(b)=c<f(a)=b##, and thus ##f(c)<f(a)=b##. It follows that ##f(f(f(a)))<b<a##.
If ##b>a##, then ##f(b)=c>f(a)=b##, and thus ##f(c)>f(a)=b##. It follows that ##f(f(f(a)))>b>a##.
I have proved that if ##f(x)## is strictly increasing, then ##\exists x\in [0,1]\,|\,f(f(f(x)))\neq x##.
Suppose that ##f(x)## is strictly decreasing, and let there be ##a\in[0,1]## such that ##f(a)=b\neq a##, then:
If ##b<a##, then ##f(b)=c>f(a)=b##.
If ##b<c<a##, then ##f(b)=c>f(c)>f(a)=b##. It follows that ##f(f(f(a)))>f(a)##
If ##a<c##, then ##f(a)=b>f(c)##. It follows that ##f(f(f(a)))<f(a)##
If ##b>a##, then ##f(b)=c<f(a)=b##.
If ##b>c>a##, then ##f(b)=c<f(c)<f(a)=b##. It follows that ##f(f(f(a)))<f(a)##
If ##a>c##, then ##f(a)=b<f(c)##. It follows that ##f(f(f(a)))>f(a)##
I have proved that if ##f(x)## is strictly decreasing, then ##\exists x\in [0,1]\,|\,f(f(f(x)))\neq x##.
I have proved that ##f(x)\neq x\implies\exists x\in [0,1]\,|\,f(f(f(x)))\neq x##!