What does $C[0,1]$ mean?

Yes it is. It is the space of all continuous functions from $[0,1]$ to $\mathbb{R}$. It has some mathematical structures under some specified operations. For example, $C[0,1]$ is a vector space over the field of reals.

In the space $C[0,1]$, points are just continuous functions. You can define operation on them like $(f+g)(x) =f(x)+g(x) = (f+g)(x)$ and multiplication like $(fg)(x)=f(x)g(x)=(fg)(x)$. These are called pointwise addition and pointwise multiplication.

$C[0,1]$ is the set of continuous functions on the closed interval $[0,1]$.