Consider minimizing x^2 when x can take values in -10 to 10 (we know the answer is 0, since we only consider real valued numbers). If we wanted to solve this problem, there are several approaches; some example approaches are: randomly try a lot of different values, set the derivative to zero, or try a cutting plane algorithm. In general, we might not know the analytical expression for the function we are trying to minimize (or it might be too complex) so we can't really find the derivative efficiently. Derivatives can also be computationally expensive to compute, so let's ignore that approach.
What we can do is to say let me find a solution for which the function is less than some threshold t, and then keep reducing t till I can't find any more solutions; this is what the article meant by finding a smaller circle inside a larger one (for each value of t, I try to find solutions that are smaller than t).
What cutting planes do is chop up the original function in to pieces - in some pieces, I know the value will be much larger than my threshold (so I don't have to search in those pieces), and in others it might be smaller - I focus on searching these pieces to find a value that is smaller than my threshold (after which I can reduce the threshold and try again). This is what (in a simplistic sense) cutting plane algorithms do; they chop up my search space.
How we select the points for chopping is crucial - bad choices (say choosing one point after another at random, or trying points very close to each other), I spend a lot of time chopping unnecessarily (or not benefiting from chopping by much). We also want to make sure our cuts really do divide the problem in to pieces which I can discard searching, and those pieces I discard should (ideally) be quite large. Until this work, the time taken to decide where to chop was n^3.373; they brought down the time to n^3 (where n is the number of variables that the function I am trying to minimize takes as inputs).
They also said that for special classes of functions, they can really improve the total computation time significantly (from n^6 to n^2).
I'm glossing over (and am certain I've got some details wrong) many issues to give a taste of the big picture idea of cutting plane approaches in general; there has been decades of work on these problems that you can read (I recommend anything written by Prof. Stephen Boyd as an introduction to some of this research).