It's quite simple. Each column represents a 'field' of knowledge: Civitas, Humanities, Power and Science. Beneath each field is a list of 'specialties' ... those for Humanities include Languages, Law, Logic and Publishing.
The mage begins with 1 field and 1 specialty in that field at 1st level. He then gains another specialty every four levels thereafter (at 5th, 9th, 13th and so on) and another field at every six levels (at 7th, 13th and 19th). This means that a 12th level mage would have two fields and three specialties.
The second table indicates the chance of knowing something in that field. Percentages for general, specific and exacting questions indicate the chance per level of the mage knowing the answer to a particular question. In addition, there are certain obvious answers - such as if the mage were to have Civitas as a field, and agriculture as a specialty, the question "What do I plant here?" would be automatic knowledge. However, if the question was, "Can I grow a plant not indiginous to this clime here?", that is a general question and the 1st level mage would have an 8% chance of knowing. The plant might still actually grow, but the mage would not know for certain until he tried.
You will note, however, that there is also a 'Researched' column. This is the chance of the mage knowing the answer to a question multiplied by the number of common books the mage possesses regarding the subject divided by 100.
Now this gets tricky. I divide my books into four categories, based on value: common, unusual, rare and arcane. One unusual book is worth 4 common books. A rare book is worth 16 common books and an arcane book is worth 64 common books. Therefore, the made need not have 100 actual books to gain the 15% bonus; if he has excellent books, he can make due with less reading.
Moreover, the mage need not actually possess the books ... he need only have access to them ... which he will have, to some extent, through the various libraries that exist, at least one of which he will know well as that it the one where he was trained to be a mage.
I suggest if this interests you that you should read further on sages in the DMG. I've used this table for some years now, and the players do pretty well with questions along the 'do I know the answer to this already' type.
I will give you an example. My party's paladin has recently decided to find a particular magic item - Horseshoes of the Zephyr. The first question is not where do we find a sage ... the first question is to ask the party's 9th level magic user, "Any idea where one might be?" The mage has not taken 'artifacts' as a specialty, but does have 'Power' as a field. The mage wasn't certain. Next question: "Any idea where we might go to find the knowledge" ... whereupon the mage thinks about where there might be a lot of books about artifacts. As it turns out, Poland. So they will go to the library in Krakow, province of Galicia, and ask for permission to look in a library there.
Frankly, I don't understand the "sages charging money for knowledge" concept at all. At no time in history has the acquisition or withholding of knowledge ever been a business strategy ... until applied by Google in the last two years, and believe me, people are PISSED. Sages charging for knowledge is just another gygaxian way for DMs to screw players. I don't subscribe to it.
Nothing left to do but to include the other tables: