It didn't limit only Chinese immigrants, and it didn't limit voluntary immigrants at all. It also was meant to block Japanese and other Asian immigrants, but only those being brought over against their will. It was primarily a means of protecting the American labour force from wages that other people couldn't compete with.

As to why East Asians were the focus in the first place, there were definitely a much larger number of Chinese coming in to America at the end of the 17th century, so the perceived threat was greater concerning Chinese immigrants, but they weren't actually the only ones singled out. Still the discrepancy in the numbers of Chinese compared to other groups becomes a lot more obvious when a few years later you have immigration outright stopped. But at least with Page, it was East Asians more generally.

Quoting from the first paragraph:

Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That in determining whether the immigration of any subject of China, Japan, or any Oriental country, to the United States, is free and voluntary, as provided by section two thousand one hundred and sixty-two of the Revised code, title "Immigration," it shall be the duty of the consul-general of consul of the United States residing at the port from which it is proposed to convey such subjects, ... to ascertain whether such immigrant has entered into a contract or agreement for a term of service within the United States, for lewd and immoral purposes; and if there be such contract of agreement, the said consul-general or consul shall not deliver the required permit or certificate.

That's the gist of Section 1. Section 2 states that anyone bringing in someone from China, Japan or any other "Oriental" country will be fined if the person being brought in has not given full consent. So no tricking people into getting on a ship and sending them to the US.

You can read the act in its entirely here.

Like I said above there were a large number of immigrants coming from China in the mid-1800s. Resentment had been building among non-Chinese in the US since the wages for Chinese workers was lower, and the growing numbers were seen as a threat to the job security of non-Chinese in the country. By the 1850s there was outright violence against the Chinese, and by the 1870s the economy wasn't doing so well, so it was sought beneficial to enact legislation to curb the numbers.

The Page Act was pretty minor compared to the Chinese Exclusion Act passed by Congress in 1882. The Page Act only worked to prevent forced immigration of East Asians. The Chinese Exclusion Act was to stop it outright, voluntary or otherwise.

edit: I just saw you posted this 1 month ago. I have no idea how I got here. I thought I came from the front page of AskHistorians, but clearly that's not the case. Anyway, I hope a late answer is better than no answer.