Rob's answer is giving the correct physics, but since you are a "smart middle schooler" I would like to address the rest of your question. To start with it will be good to increase your background on axions by reading the wikipedia article. , for encyclopedic information.
Axions are the particles that let energy flash between a matter particle and its anti-particle.
In particle physics very strict mathematical rules describe the interactions between particles. Particles are on the same status as antiparticles and nothing particular "flashes" unless there is an interaction. Interaction means energy and momentum is exchanged. Usually particles and antiparticles annihilate when they interact, and a plethora of Feynman diagrams describes pictorially the mathematics of their interaction.
But according to quantum field theory, every particle has an anti-particle (besides the majorana fermion).
Particles can be antiparticles of themselves, as for example the photon. In general bosons are antiparticles of themselves if they are neutral, and the axion is neutral.
So if axions have an anti-particle, why don't they react?
Interactions happen with energy and momentum exchanges between particles, and theory up to now can predict the probability of various interaction products coming out using field theory and the Feynman diagrams. The existence of antiparticles does not have anything to do with interactions, which need energy and momentum to happen.
In some theories, if the axion has mass it can decay into photons , the energy supplied by its mass.
This is a Feynman diagram for an axion interaction with a magnetic field, that can allow calculating the probability of this happening within a specific theoretical model:
Here is the decay, where the mass of the axion provides the energy.
One can then run an experiment trying to detect photons that could be attributed to axion interactions and decays.
Maybe your flashing statement refers to vacuum loop feynman diagrams . Those diagrams cannot exist by themselves, they need an input energy and momentum, they are higher order (much smaller probability) Feynman diagrams that have to be added to calculations to make them more accurate.
A lot of study in mathematics and physics is needed to be able to understand the present model of particle physics interactions.