"Keeping up with" something means doing it quickly enough and often enough. The opposite of "keeping up" is "falling behind".
If you're running with someone, "keeping up" means that you run fast enough to stay with other people. In the example at top, the teacher wants her students to read a little every day or week. That's because, if they wait until too late, there will be too much left for them to read and they will "fall behind".
You can also talk about "keeping up with" news and with people who you used to know: