What does the President being “on the line with China” mean?

In this context “on the line” means “on the telephone”.

It’s not used entirely literally – as you said yourself in your comment it’s unlikely that one President would call another to ask to borrow money directly.

Mr. Romney’s main thrust is an economic one – he is contending that President Obama is ignoring those Americans who favour the government borrowing less because his administration is busy establishing policies that require the government to borrow more. He singles out China (which, as Brandon explains in his answer, happens to be the source of a large proportion of that borrowed money) in order to appeal to voters uncomfortable with the growing economic influence of a country that the U.S. has traditionally held as an adversary.

Romney is intentionally pandering to a perception that may be commonly held by elements of the American public that have little formal training in economics.

This comment is fallacious or, at the very least, misleading. The US government does not borrow money directly from other countries - the government, through the treasury, issues treasury bonds and other "government debt securities" in return for a promise to pay the face value plus interest (you hear this in the news as "bond yields"). When the government (technically the treasury) sells bonds, anyone can buy them. It just happens to be that China (and private investors within China) buys a significant proportion.

But as you can see from the link below, China is not the only country to own US government debt. There's a good description and overview of ownership of US debt securities, here: http://www.gao.gov/special.pubs/longterm/debt/ownership.html