Why was the Alexandrian library burned down? How much effort did it take, as I heard it was fire-prove?
First of all, it is important to note that the Museum and Library were two different institutions, often collectively called The Library at Alexandria today. The former hosted scholars, the latter the institution collecting the books. It is also important that the library probably had collections in more than one place in Alexandria - the Serapeum is often pointed out as a secondary location.
When the library at Alexandria was destroyed is actually unknown and modern historians tend to talk about it as a gradual process rather than event (burning). As for an example, Caesar is often blamed in popular culture, but we know that the library was active for centuries after his siege ended 47 BC: Claudius was interested in the Museum and expanded it, and Caracalla removed the income for the museum another 200 years later. In 272 AD the emperor Aurelian reconquered Alexandria and the area Burcheiron in Alexandria was destroyed, presumable with large parts of the remains of the library and museum.
Here are the relevant written sources relating to Cesars siege and the burning of the library at that time. What I want you to note is how the narrative change over time. The earlier (contemporary or near contemporary) do not mention a destruction, or just limited damage. Then again, they might be biased because the men in power at this time did not want the be connected to the destruction of the library.
Caesar - De Bello Civili 3.111 (written between 49 and 44 BC)
Achillas trusting to the valour of his troops, and despising the handful of men that followed Caesar, quickly made himself master of Alexandria, the palace only excepted, where Caesar thought proper to make his stand, and which he attacked briskly, though without effect. But it was on the side of the harbour that the greatest efforts were made. On that, in effect, the victory depended. Besides two and twenty constant guard-ships, there were in the port fifty galleys, from three to five banks of oars, which the year before had been sent to Pompey's assistance, and were returned since the battle of Pharsalia. Had Achillas been once master of these vessels, he might have cut Caesar off from all communication with the ocean, and consequently from all hopes of receiving supplies of victuals or forces. Thus the Egyptians, in hopes of a complete victory, and the Romans to avoid a certain ruin, exerted themselves with incredible vigour. At length Caesar carried his point, and not only set fire to the vessels abovementioned, but to all that were in the arsenals, after which he passed some troops into the Isle of Pharos.
Lucanus – Pharsalia X.606-608 (written 61-65 AD)
Nor fell the blaze upon the ships alone, But seized with writhing tongues the neighbouring homes, And fanned to fury by the Southern breeze
Seneca – De animi tranqullitate IX.4 (dead 65 AD)
The inner curve is the one to take, not only in running races and in the contests of the circus, but also in the race of life; even literary pursuits, the most becoming thing for a gentleman to spend money upon, are only justifiable as long as they are kept within bounds. What is the use of possessing numberless books and libraries, whose titles their owner can hardly read through in a lifetime? A student is over-whelmed by such a mass, not instructed, and it is much better to devote yourself to a few writers than to skim through many. Forty thousand books were burned at Alexandria: some would have praised this library as a most noble memorial of royal wealth, like Titus Livius, who says that it was "a splendid result of the taste and attentive care of the kings.
Plutarchos – Parallel lives: Caesar 49 (dead c. 120 AD)
When the enemy tried to cut off his fleet, he was forced to repel the danger by using fire, and this spread from the dockyards and destroyed the great library
Aulus Gellius - Attic nights VII.17.3 (dead c. 180 AD)
At a later time an enormous quantity of books, nearly seven hundred thousand volumes, was either acquired or written in Egypt under the kings known as Ptolemies; but these were all burned during the sack of the city in our first war with Alexandria, not intentionally or by anyone's order, but accidentally by the auxiliary soldiers.
Dio Cassius – Roman history 42.38 (dead c. 235 AD)
After this many battles occurred between the two forces both by day and by night, and many places were set on fire, with the result that the docks and the storehouses of grain among other buildings were burned, and also the library, whose volumes, it is said, were of the greatest number and excellence
For some modern sources see:
Edit: spelling