Why the colour of KMnO4 at the end point of titration may disappear after some time?
The most likely reason for the colour of permanganate to vanish after a few seconds even though you thought you have the end point is: you have not reached the end point.

Acid/base reactions are typically extremely fast reactions; their rate is not even limited by the speed of diffusion since protons can be shuffled along via the Grotthuss mechanism much more rapidly. Thus, you can assume each acid/base reaction to instantly reach completion.

With practically all other reactions that is not the case. They are typically slower and can be much slower. The reacting species must meet each other in solution which can take time. At the beginning of the titration, there are still many iron(II) ions floating around ready to be oxidised by any incoming permanganate. Towards the end, iron(II) is getting more scarce. So a permanganate ion may linger around much longer before it meets an iron ion. You should give it a certain time in case it does find one. The value of $\pu{30s}$ is arbitrary. Ten seconds or two minutes would probably do equally well (except that ten seconds may or may not be slightly on the short side). The book/method just needs to present any ballpark value.

So the reaction occuring that causes the discolouring of the solution is indeed exactly the reaction you are observing: the oxidation of iron(II). It is merely slowed down so it can take a few seconds to proceed to completion when close to the equivalence point.