When Krugman talks about savings, he is presumably talking about the difference between production and consumption (personal consumption and government consumption) in a country as a whole. Although measured in monetary terms, this is really about stuff, i.e. goods and services. It is not about personal savings of money in a bank.
Some of the difference will be used to make things producing stuff for the future, i.e. as investment. In a closed economy, in this sense savings must equal investment, since building up stocks of stuff for future use would also count as investment, while simply wasting the excess stuff would make it not count as valuable production.
In an open economy, it is possible for production and use (consumption and investment) to differ, and so it is possible for savings and investment to differ. If use exceeds production, i.e. investment exceeds saving, the extra stuff has to come from somewhere, namely as net imports of stuff. If production exceeds use, i.e. saving exceeds investment, then the extra stuff has to go somewhere, namely as net exports.
So what Krugman is saying is that having a trade deficit is associated with production being less than use, and so with savings in this sense being less than investment. So he is suggesting that the US trade deficit is associated with low savings. Using his definitions, that is almost tautologically true as a matter of arithmetic. It is also more coherent than Trump's simple exports good, imports bad, which is little more than saying send stuff abroad but do not get stuff back.
Krugman's next step is to say that the US trade deficit is a consequence of low savings. That is putting a suggestion of causality into the relationships, something much less obvious. It also raises the question of how to change the position; elsewhere it appears that Krugman would not support efforts to reduce consumption or investment. He would welcome increased production; so too would Trump.