The other answers already provide some very interesting insights on the matter, which largely amount to the fact that the movie is more or less intentionally left ambiguous and without a clear interpretation, which is also what makes it so fascinating and engaging.
I'd like to amend this here with Wolfgang Schmitt's interesting analysis of the movie (unfortunately not available in English). He goes down pretty much the same line and concludes that the movie in its lack of a clear explanation is basically a statement about the universality, non-representationality and non-explainability of art itself. He says that the movie is actually deliberately constructed in a way to prevent any secondary interpretations of it and to prevent an easy-out approach to explaining it (translated by me):
...Why is the structure of 2001 so complicated? That is because it is actually layed out that way, it's constructed in a way so that it can't be explained with one working interpretation. [...] Every narrative thread we follow is lost in the void at some point. Even the episode about HAL where we think: "yes, that's quite tangible", we can say we have artificial intelligence and human intelligence and one will prevail over the other. Even if we realize this, the movie stops at this point and continues at a totally different point. Stanley Kubrick intentionally wants to lock himself from the secondary [...] He is concerned about the primary. And he wants to lock himself from something else, maybe even stronger than from those interpretations: from the nowadays so popular arts education that thinks you just need to take a work from its pedestal a little and suddenly you can serve it to everyone, that thinks art is democratic, you have to "pick up the people where they stand". Exactly that is what Kubrick does not do. He locks himself and that's what makes the greatness of his work.
I think it's great that especially this movie is so popular, although it stands against all that what we generally expect from art nowadays. Especially we always think that the artwork has to communicate with the viewer [...] But Kubrick's work does not communicate, and that's already evidenced by the fact that there isn't hardly spoken a word in this movie [...] Because A Space Odyssey is, first and foremost, a movie about art, as there is a very radical artist standpoint assumed here. 2001, that's actually the manifest of Kubrick.
For this, he chooses the monolith as an example of a completely uncontextualizable and unexplainable piece of art but that still provokes a change, equating this to the 20th century art movements of suprematism and minimalism that understood art as only having itself as content:
Let's take a look at the most mysterious thing in the whole movie, the black monolith. What does it stand for? It appears in all spaces and all times over and over, and there's always change induced by its mere presence [...] But let's think about the art of the 20th century. Don't we find there also such art objects like this black monolith? When we think about the suprematism, about Kazimir Malevich and the "Black Square". That's basically beginning and end of art in one, the "Black Square" stands for the art, it seems to be the concentrate, it's concrete, it's a direct experience when he stand in front of this "Black Square", and at the same time we can't say for what it actually stands. Or let's think, for the movie is from 1968, about the minimalism which is especially popular in America [...] There we find such sculptures that are just there in the room but behind which there isn't a direct meaning so we can't say "this stands for that" like we know it from classical paintings. [...] When there is something in the room, we can usually contextualize it [...] But what when there's something that at first doesn't have any meaning and no connections to the other things in the room? Then we're really confused, our coordinate system doesn't work any more. That's what 2001 is about. The monolith is a work of art that locks itself from every interpretation and especially every education. And the closedness is grounded in the fact that the artwork only relates to itself. There is a nice quote from Kazimir Malevich and his thoughts about art:
The new art has placed the principle into foreground that art can only have itself as content. So we don't find the idea of something in it, but only the idea of art itself, its self-content. The inherent idea of art is the non-representationality.
Kubrick follows exactly this tradition.
And when now equating the movie itself to this monolith he says, that the movie demonstrates the universality and power of art that actually comes from its unexplainability and that the fact that we'll never completely grasp the movie is its advantage and gives it it's actual meaning.
We stand in front of Kubrick's movie like in front of the black monolith. [...] This monolith seems to change something. We can approach it, we can try to explain it, but we never really grasp it completely. There's hardly a movie that supports such a radical concept of art and that's so popular at the same time. Therein lies the art of Stanley Kubrick.
The film starts and ends with music but in a quite strange way, for before we see the intro [...] there is just nothing and we hear music [...] And also at the end it's the same, when the end credits are all over we still hear Johann Strauß' waltz melodies. What does this actually mean? When we look at the screen we actually see nothing, everything is black [...] One could also say the monolith was zoomed comletely close here, and from this monolith, this auratic work of art, everything is possible [...] So the music at the beginning and the end isn't a prelude and afterplay, but it's an approach to art that says "art is always there". [...] With 2001 Stanley Kubrick has created a movie about cinema itself. For from this place where you first see nothing everything is possible. And where does film start and where are its limits? Do we need a clear plot structure? No, that's clear from this movie. Do we even need images? Even that is questionable, alone the fact that there is something acoustical and we look in the direction of the screen already means we encounter an artwork [...] Kubrick said about this movie, and he didn't say much about his movies, "In 2001 the message is the medium itself", so it's about the movie itself. 2001 itself is the monolith, an erratic work of art we can only approach but never completely understand, but this work or art changes us. And that's what great art can do.