There are several options -- which makes sense, as this is a story which explores the idea of infinity.
The Library is spherical
Immediately following the line "I say that the Library is unending", the narrator says:
Let it suffice now for me to repeat the classic dictum: The Library is a sphere whose exact center is any one of its hexagons and whose circumference is inaccessible.
This is one way the Library can be infinite -- if the Library wraps around and re-connects to itself, it is infinite, in the sense that you can walk in a single direction forever. The number of hexagons and books are finite, but the Library itself has no beginning and no end.
The Library repeats infinitely
The story concludes with this thesis:
I say that it is not illogical to think that the world is infinite. Those who judge it to be limited postulate that in remote places the corridors and stairways and hexagons can conceivably come to an end -- which is absurd. Those who imagine it to be without limit forget that the possible number of books does have such a limit. I venture to suggest this solution to the ancient problem: The Library is unlimited and cyclical. If an eternal traveler were to cross it in any direction, after centuries he would see that the same volumes were repeated in the same disorder (which, thus repeated, would be an order: the Order). My solitude is gladdened by this elegant hope.
In other words, he posits that the Library is literally infinite; that the 25^1312000 individual books repeat over and over.
He calls this "elegant," as yet another hope for order in random chaos -- here is suggesting, not only that there are infinite copies of all books, but that the arrangement of those books is preserved. This is an order that cannot be perceived, tested, or comprehended by any mortal, but its (hypothetical) existence is a comfort to the narrator.
Another variation arises naturally -- that the Library does extend infinitely, but with different arrangements of the books. Eventually, of course, you would exhaust the possible arrangements of books -- but you could create new configurations of those arrangements, and their location respective to one another.
However you look at this, Borges is demonstrating the idea of magnitudes of infinity - e.g. the idea of an infinity of infinities.
The Library is metaphorically infinite
Part of the point here is that the narrator has no idea what the structure of the universe is. He has no way to discover it. He can guess and conjecture, but is overwhelmed by the ungraspable nature of the Library, and of infinity.
The thing is, infinity is not a concept the human mind can grasp easily. In many ways, it is easier for us to understand something finite but whose immensity presses upon us (i.e., a finite Borgesian Library), than the infinity of our own universe (where it is so easy to focus on our own planet, country, home).
Understanding true infinity is very difficult; creating an example of true infinity might have resulted in less powerful imagery. But Borges's Library is immense in a way we can grasp intuitively. It gives you a sense of scale -- the difficulty of finding a single comprehensible word in a book generated from random characters; the existence of books that are works of truth and art, as the product of randomly combining characters until one forms. This is not literally infinite, but it feels so near the real thing, as to be metaphorically infinite, to demonstrate the sense of infinity even if it isn't the real thing.