Broad construction, or loose construction, has to do with interpreting the Constitution. It is the opposite of strict construction.
The issue here is how to read the Constitution. Strict constructionists in the 18th century argued that the government should only be able to do what the Constitution explicitly said that it could do. If the Constitution did not say that Congress could make a bank, then Congress could not make a bank. But broad constructionists looked at it the other way. They said that government should be able to do anything as long as the Constitution did not forbid it. Congress could make a bank as long as the Constitution did not say that it could not.
The broad constructionists won this battle. The major case having to do with this was McCulloch v. Maryland.