How to have productive meetings with Ph.D supervisor
Speaking from experience, both as a student and as a Phd supervisor in theoretical computer science. The meetings with your supervisor serve many purposes. To narrow the scope a little bit, let me only discuss the type of meetings you mention - where you read a research paper and discuss it to a high level of detail with your advisor. 

In TCS (and maths, I would assume), the primary goals of having the student read a paper and present it to the supervisor are:


So that the student learns the contents of the paper, especially the main technical tools used and developed in the paper, to a degree where the student can recognize a setting where these tools can be applied, and apply the tools in relevant settings. Because this is research the tools will have to be modified at least slightly every time they are applied. Hence, in order for the endeavor (reading the paper) to be any useful at all the student needs to achieve a very deep understanding of the paper she is reading. 
So that the student learns to read, understand, and present (and eventually write) technical proofs. This is a big one - it is incredibly important that whenever you feel that you understand a proof, that you are able to present it at various levels of detail. On one end you should be able to give a “big picture” view of what the Lemma statement actually means, what it gives you, why it should be true, and why this is a non-trivial statement. It is equally important that when probed, you are able to re-create to full detail a presentation of the proof. By full detail I really mean full detail - are you saying “let x be an element of S” you better have proved that S is non-empty first. Are you invoking Lemma 3.5? You have to point to precisely which objects you are applying Lemma 3.5 to, why these objects satisfy the premise of Lemma 3.5, and which objects this invocation of Lemma 3.5 generates for you to use further in the proof.  


The level of understanding that I describe above is really the only level that will be useful for you in your research. It also takes a lot of work to get there, this goes way beyond the level of understanding you need to get A’s in your classes. For this reason it is perfectly normal to take 2 months to read a 10 page paper. This is also a great cause of misunderstandings and frustration between students and advisors (the student thinks “I already know this to a level where I would ace an exam on this paper”, while the advisor thinks “the student has no idea what is going on”). 

I can not speak for your advisor, but in such discussions with students I often ask follow up questions in order for the student to herself come to the realization that she needs to understand this or that part of the proof better.

Also, professors are not gods. In fact, for me, the vast majority of follow up questions that I ask is because I am struggling with understanding what it is the student is talking about, and I’m trying desperately to figure out whether I missed something obvious, or whether the student is talking nonsense. Doing this (for 2 hours) is mentally exhausting to the point of the infeasible. For this reason the conversation will often naturally after an hour or so shift to lighter topics. 

I can’t actually know whether any of the above applies to you, but I do find that the above is a pretty common problem. Let me end with some pieces of advice (much of which has already been said in other answers).


Talk to your supervisor about this. This might well not bring about the change you are looking for, but you will better understand each other’s motivations, frustrations, etc.
If you are worried about your productivity, show initiative. Ask your supervisor about what are the next natural papers to read, with or without discussing these papers with her. Then read them, to the best of your abilities. Ask your supervisor for research questions, and think about them. Talk to your fellow students, and initiate research projects with them. Remember that in the last year or two of your Phd. you should be able to complete a research project from A to Z without any help from your supervisor at all. This involves obtaining the relevant background, asking feasible and interesting research questions and answering them. The best way of getting there is to try to do this as much as possible from day 1 of your Phd studies, and then using your supervisor to help you with your weakenesses. As an example, when I was a student I had no problems with solving problems, but had no idea how to find relevant research questions, and so I would lean on my supervisor and collaborators for this to a large degree. For you there might be another aspect of research that you will need extra support for, and that is what supervisors are for!