Explaining the status of the kohen, the Torah says, V’kiddashto – “You shall revere him” (Lev. 21:8). The root of the word is the same as Kiddush – the declaration that Shabbat is holy. In the case of the kohen, we have to show in thought, word and speech that he has a holy status, and when it comes to the Torah reading we place him first.
The alternative might be that people would scramble for the right to be called up at the top of the list, and there would be unseemly rivalry. Having a set rule that the kohen comes first, followed by the levi, makes things easier.
Unfortunately it does not entirely eradicate the problem, because there can still be unpleasant jostling for the third position, i.e. the first slot available for an ordinary Yisra’el. “I am more important than he is!” is the indignant protestation.
The question that arises is how we are meant to judge people’s respective level of importance. It is highly likely that “I am more important” means, “I have more money” (do you remember the song, “In this world one thing counts – in the bank large amounts!”)
Halachic sources endeavour to handle the issue by a list of religious priorities, regarding as more noteworthy a person who is marking a specified event in their religious life cycle. It doesn’t always work, but it helps. Of course the truly worthy person is not interested in outward show at all and does not push himself forward to be called to the Torah third or to receive any other honour.
There is a rabbinic saying (actually not in the Talmud, but quite Talmudic in its thinking), “He who runs after honour, honour runs away from him”.