How to understand "stovepipes" in the following sentence:"In combining these responsibilities under one overarching concept, homeland security breaks down longstanding stovepipes of activity that have been and could still be exploited by those seeking to cause harm."

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I have not heard of the term “Stovepiping” before but I looked it up and hopefully can simplify the term so it is understandable: it is a metaphor, basically it means that information or activity is focused on one purpose and ignores the context. Stovepiping has a negative connotation.
In this case, the agency (Homeland Security) is trying to get rid of Stovepiping activities so that everyone works together on a common purpose. If people work together, and they know the activities other people in Homeland Security are doing, then they can work together with better understanding. If they do not share information, then they could miss a connection between information/threats and that could be a security risk.

A stovepipe is literally a pipe in a stove. This is for very very old stoves, before gas and electricity, when a fire was lit in the stove. Because there was a fire, there was smoke. That smoke was directed out of the stove in a pipe. The smoke did not spread through the room/kitchen, the stove pipe directed the smoke directly outside.

In the metaphor here, the activity is the smoke, the activity is not seen or heard by outsiders. So while a stovepipe is good for a stove, it is not a good process for people or organizations where it is good to work together and share information :)