What is the mathematical meaning of the 'construction'?

It is a construction because $\mathbb R$ has been constructed before.

A full construction of the real interval would look like this (note that I only sketch this construction):

We start with constructing the natural numbers one by one by simply collecting all the numbers we have already constructed.

Initially we have constructed nothing, therefore the first number we construct, $0$, is given by the empty set (the set of nothing). So we have

$0 = \emptyset$.

In the next step, we construct the next number, $1$. The only number we've constructed so far is $0$, so we get

$1 = \{0\} = \{\emptyset\}$.

Then we construct $2$, again as the set of numbers defined so far. So we get

$2 = \{0,1\} = \{\emptyset,\{\emptyset\}\}$.

You see how this works. Note that set theory guarantees us that all those sets exist.

Next we collect all those numbers into one set,

$\mathbb N = \{0,1,2,3,\ldots\}$.

This is a non-trivial step; set theory indeed contains a separate axiom just to say that this is indeed a set.

We also define addition, multiplication and order on $\mathbb N$, mainly by following Peano's axioms.

Next we construct the integers as differences of natural numbers; formally this defines them as equivalence classes of pairs of natural numbers with certain rules for order and operations. Pairs, in turn, can again be defined through sets, for example as

$(a,b) = \{\{a\},\{a,b\}\}$.

Then we define the rational numbers as quotients of integers (again, formally as equivalence classes of pairs).

Next we define the real numbers $\mathbb R$; there are several possible constructions, but lets assume that we construct them as Dedekind cuts. That is, a real number is a downward-closed non-empty proper subset of the rational numbers that has no maximal element. And again, we have to define order and operations.

And now that we have $\mathbb R$, we can define the interval $[0,1]$ as

$[0,1] = \{x\in\mathbb R: 0\leq x\leq 1\}$.

Note that if you follow back all the constructions used, you ultimately find that the interval $[0,1]$ is constructed through sets that contain other sets that contain yet other sets, but ultimately you'll always sooner or later end up at the empty set. So we really have constructed the interval out of “nothing” by recursively collecting this “nothing” into ever more complicated sets.

And I think it is now clear why normally one stops at something that has already been constructed earlier, instead of always writing down the full construction from scratch. If you would flesh out the construction I sketched, complete with proofs that each of the steps really is allowed, and that each step really results in something having the structure we want (e.g. that the real numbers such constructed indeed fulfill the axioms of a complete ordered field), I think you could easily fill up a complete book.

A construction is an explicit (possibly complex) sequence of steps that can be followed to define an object from objects that are already available. The construction can involve infinitely many objects. In your example, you have the set of all real numbers and use it to create a new object (the closed interval from $0$ to $1$). Similarly, we say that we construct the natural numbers from sets, the integers from the natural numbers, the rational numbers from the integers, the real numbers from the rational numbers and the complex numbers from the real numbers.

A statement of the form "there exists an object with property $P$" is non-constructive. However, it is sometimes possible to convert such non-constructive statements into constructions by showing a way to explicitly construct a particular object with property $P$.