What is behind of the attraction between a burned matchstick with a magnet? I do not know how to understand the reasons of why a burned matchstick is attracted by a magnet (most probably) of Neodymium. 

Here are some screenshots of a video in which this natural phenomenon is presented:







(In the video, see at 1 min 25 s):


Source
Source
Some matchstick heads contain iron(III) oxide as a colorant.



The yellowish color of the burning match indicates that it has low oxygen, i.e. a reducing flame. It reduces the iron oxide to iron which is attracted by the magnet.

The reduction reactions that occur are probably quite complex. Below are two simplified possible reaction equations. $\ce{(CH2)_n}$ in equation (2) stands for paraffin from the splint's impregnation. 

$$\ce{Fe2O3 + 3 CO -&gt; 2 Fe + 3 CO2} \tag{1}$$

$$\ce{Fe2O3 + \frac{1}{n} (CH2)_n -&gt; 2 Fe + CO2 + H2O} \tag{2}$$