In matter-antimatter interactions, what's the meaning of "touch"?

At the level of the standard model of particle physics there is no "touching". There are interactions which transfer four momentum between particles , defined strictly within the model and represented with quantum field theory feynman diagrams. These diagrams allow one to compute crossections ( i.e.how large does each particle appear to the other) and decays, and the values depend on the energy and momentum of the particles.

This is part of a feynman diagram

What does it mean to say that particles "touch"? Is it the same kind of "touch" that happens when my hand's motion is stopped by a wall?

At the particle level, when your hand touches the wall, there are very many interactions between the electrons in the atoms of your hand and the electrons in the atoms of the wall, all governed by the exchange of virtual photons,but this you will be able to understand when you take a course in quantum mechanics.

This is how electrons interact with electrons

The electrons repel each other and you feel it as "touch". Keep in mind that the atoms in matter are of the order of $10^{23}$ per mole, both in your hand and in the wall.