Let's start with who, shall we? Suetonius, Tacitus and Cassius Dio are all historians who claim that Nero started the fire, although the way he did so varies - as do the reasons.

Tacitus was a senator and consul during Nerva's reign, he writes:

A disaster followed, whether accidental or treacherously contrived by the emperor, is uncertain, as authors have given both accounts, worse, however, and more dreadful than any which have ever happened to this city by the violence of fire. [---] (Tac. Ann. 15.38)

Nero at this time was at Antium, and did not return to Rome until the fire approached his house. [---] These acts, though popular, produced no effect, since a rumour had gone forth everywhere that, at the very time when the city was in flames, the emperor appeared on a private stage and sang of the destruction of Troy, comparing present misfortunes with the calamities of antiquity. (Tac. Ann. 15.39)

Suetonius was a sort of staff member under Trajan and thus close to the imperial power, he writes that:

He spared, moreover, neither the people of Rome, nor the capital of the country. Somebody in conversation saying: “When I am dead let fire devour the world." "Nay," said he, "let it be while I am living." And he acted accordingly; for, pretending to be disgusted with the old buildings, and the streets, he set the city on fire so openly, that many of consular rank caught his own household servants on their property with tow, and torches in their hands, but durst not meddle with them. [---] This fire he beheld from a tower in the house of Maecenas, and, "being greatly delighted," as he said, "with the beautiful effects of the conflagration," he sung a poem on the ruin of Troy, in the tragic dress he used on the stage. [---] (Suet. Nero 38)

Cassius Dio is writing almost two centuries after the end of Nero's reign so his retelling is questionable already for that reason, but he mentions a small detail that might make him interesting to add here:

After this Nero set his heart on accomplishing what had doubtless always been his desire, namely to make an end of the whole city and realm during his lifetime. At all events, he, like others before him, used to call Priam wonderfully fortunate in that he had seen his country and his throne destroyed together. [---] (Cass. Dio. LXII, 16)

The why of your question should thus be quite clear. Historians in previous centuries often took the primary sources literally. /u/Daeres actually made a post about this way of writing history over in /r/BadHistory, if you're interested. Anyway, Suetonius commonly exaggerates these histories about emperors (see Tacitus' statement below) so it comes as no surprise that he blatantly states that Nero set fire to Rome because of selfish reasons and then proceeded to sing as song about it. As I wrote, Suetonius was a person who stood close to the imperial Roman power. As such, there was an example set by Augustus that essentially was for the emperor to always act as iconically as possible. As the pater patriae he was. However, while this is the stance Suetonius (and likely most of the senators and such in Rome), that is not the people's opinion of Nero. He was actually rather liked with the people, there is even a quote about him that I read recently that says something along the lines of: "Octavian had to try hard be a popular figure with the people; Nero didn't have to try, it came naturally to him".^1 If you look at the other two historians I presented, one of them - Cassius Dio - I've already said, writes almost 200 years afterwards, so he's actually writing based on historians like Suetonius and Tacitus (considering what he writes, it's likely that he writes mainly off Suetonius in this case). Tacitus, as you can see, doesn't actually say that Nero started the fire, he merely implies it. So in essence, the main reason Nero has been accused of starting the fire even in modern times is because it took until the 20th century before people questioned it properly.

Ironically, Tacitus states in the beginning of his Annales that:

The histories of Tiberius, Caius, Claudius, and Nero, while they were in power, were falsified through terror, and after their death were written under the irritation of a recent hatred. Hence my purpose is to relate a few facts about Augustus—more particularly his last acts, then the reign of Tiberius, and all which follows, without either bitterness or partiality, from any motives to which I am far removed. (Tac. Ann. 1.1)

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[1] If it's deadly necessary I can make an effort to find the source, but I don't remember exactly where I read it.