What he really means is that branches are now a part of your daily workflow. You can sync up to head (or wherever you pulled your working copy of the repository from), create a branch, and start working on some feature on that branch, mucking up a hundred files along the way. Now say some some urgent issue comes up that you need to fix immediately -- you can easily go back to the point before you branched, and create a new branch from there on which you can fix the issue. Meanwhile, your previous branch still exists, so when the urgent issue is fixed you can resume where you are working on. Another great thing about branches is say you're working on some feature in a branch and you come to a point where you can implement something multiple ways, but don't know which one will turn out elegant. You can create a new branch from that point in your current branch, and if it doesn't pan out, revert it to where you were.

The kicker with Git is that all these branch operations take on the order of milliseconds because Git stores the entire project history -- all past revisions of files, etc -- locally. Sure, you pay in a bit of disk space, but consequently almost all common operations can be done without hitting the network and are crazy fast. Even when you execute a commit, you're not committing to a server, but to your working copy of the repository. Later, you can push your changes somewhere else and merge accordingly.