Why is energy being released when an electron is added to a neutral atom?

The force between two charges #q_1# and #q_2# separated by a distance #r# is given by:

#F=(q_1q_2)/(4πepsilon_@r^2#

Where #epsilon_@# is the permittivity of free space and is a constant.

If we separate one of the charges from #r# to infinity the work done is given by:

#W=int_r^ooF.dr#

So:

#W=int_r^oo(q_1q_2)/(4πepsilon_@r^2).dr#

# W=-(q_1q_2)/(4piepsilon_0r)#

Similarly if a charge #q_1# (in this case the electron) is brought in from infinity to an energy level in an atom at a distance #r# from the nucleus then this energy would be released, assuming #q_2# is equal to the effective nuclear charge.

How would this energy appear? As the electron crashes into the atom its kinetic energy would be transferred so would appear as heat.

Since this is also an electron transition from #n=oo#, photons would also be given off.

All 1st electron affinities are exothermic for those that have been measured.

You can refer to the table below. Even sodium has an exothermic value.

https://en.wikipedia.org/wiki/Electron_affinity_(data_page)

The video below shows this happening in the reaction between iron and fluorine but bear in mind that the electron affinity is only one step in the whole energy cycle of forming an ionic compound from its elements.

http://www.periodicvideos.com/videos/009.htm

There is no reason that an electron "doesn't belong in a neutral atom". On the contrary, this is the normal state for many elements.
For example chlorine exists exclusively in nature as a negative ion.