The decision did not assert or assume this. It rested on the idea that individual have rights, and they do not lose those rights by acting collectively.
If I'm allowed to speak, then I'm allowed to buy and use a printing press so that my speech can be more easily distributed. If I can't afford a printing press, I can go in halves with you, so that we can both print off our pamphlets. If neither one of us can afford a printing press, we can take up a subscription among our friends to buy one.
Or to put it another way: The New York Times does not have a free speech right, but the humans who work at and own the New York Times do, and gagging the New York Times would infringe on the rights of those humans. Any law that implies the New York Times can be gagged (eg, because it's a corporation, and thus has no rights) is clearly wrong.
I assume you would NOT be okay with that, but keep in mind that in the US, the first amendment applies to us as individuals. If those rights are forfeit when exercised collectively, then the result is a world where only the very, very rich can speak without the permission of the government. That's a horrifying outcome!
The rich, I would argue, have too much power already, and stripping the ability of the non-rich to band together to act as a counterbalance does nothing more than cement that power.
And corporations are not just people acting collectively, they are financial instruments that incentivize certain behavior.
I think the main problem with the decision was the assertion that entities like corporations, rather than simply individuals, have free speech.