Much of the alien's purposes were explained in chapter 20. To pick out a few of the relevant points:
- They wanted information they couldn't get just from distant observation:
"Don't think of us as some interstellar sheriff gunning down outlaw civilizations. Think of us more as the Office of the Galactic Census. We collect information."
"There was so much your television programs didn't tell us. Oh, we could figure out your technological level pretty well, and a lot more about you. But there's so much more to your species than that, things we couldn't possibly learn indirectly."
- They wanted to help, something else that probably couldn't be achieved effectively without some level of direct contact:
"The [Hitler broadcast] was alarming. We could tell you were in deep trouble. But the music told us something else. The Beethoven told us there was hope. Marginal cases are our speciality. We thought you could use a little help."
- Also, the Machine wasn't just a one-time deal: it allowed them to set up a permanent transit system to Earth, giving them the ability to observe more directly and, if necessary, intervene directly, though they considered that scenario unlikely:
"We think they did almost all the work." Vaygay was explaining his and Edna's theory on what the five of them had experienced. "All the project did was to make the faintest pucker in space-time, so they would have something to hook their tunnel onto."
"The Earth is linked up now, right? Both ways. If we can go home, you can come down to us in a jiffy. You know, that makes me awfully nervous. Why don't you just sever the link? We'll take it from here."
"Sorry, Presh." he replied, [...] "But we don't expect to use it."
They could arrive almost instantaneously, perhaps only in Hokkaido, perhaps anywhere on Earth. It was a transition to what Hadden had called microintervention. No matter what assurances they gave, they would watch us more closely now.
... but there's more to it than that. If all they'd wanted was to tell us about π, there would have been no need for the Machine, but by the same token, if all they wanted was for us to build the Machine, there'd have been no need for the Message to be so heavily obfuscated. This is one of those cases where it isn't so much about the destination as the journey. This is never explained in detail; the author deliberately leaves it as one more mystery. But it is heavily implied that it is all part of the alien's purpose, to help out a marginal but promising young civilization.
(Part of the purpose, however, may have been simply to attract attention. The drama around the Message, the controversy about whether or not to build the Machine, did undeniably attract the attention of all mankind, in a way that a message about transcendental numbers would have been unlikely to achieve.)