Do protons exchange photons with electrons? I'm sorry for this question but, I just don't get it. According to the electromagnetic field theory, electrons repel each other by exchanging photons. How do protons attract electrons, by photon exchange?
 A: ELECTRON-PROTON ATTRACTION: a simple, semi-classical  analysis to avoid full scale QFT.
The exchange of the photon between the proton and the electron leads to attraction, only because the total energy of the electron is negative. 
Let us consider the hydrogen atom for simplicity, and imagine the funnel-like shape of the electron energy.  The total energy of the electron in the hydrogen atom at distance $r_0$ from the proton is 
$E(r_0)=-\frac{e^2}{8\pi\epsilon_0 r_0}$.
If the distance $r_0$ is sufficiently short, then the electron will emit a photon which will be absorbed by the proton, and the amount of energy of the exchanged photon will be dictated by the uncertainty principle: 
$\Delta E\Delta t=\hbar.$ 
But $\Delta t=\frac{r_0}{c}$ so that  
$\Delta E r_0=\hbar c\rightarrow \Delta E= \frac{\hbar c}{r_0}$ 
So the new energy of the electron will be
$E_1=-\frac{e^2}{8\pi\epsilon_0 r_0}-\frac{\hbar c}{r_0}=-\frac{e^2+8\pi\epsilon_0\hbar c}{8\pi\epsilon_0r_0}$
or the equivalent amount of energy corresponding to some new position $r_1$
$-\frac{e^2}{8\pi\epsilon_0 r_1}=-\frac{e^2+8\pi\epsilon_0\hbar c}{8\pi\epsilon_0 r_0}$
from which we get $r_1$ in terms of $r_0$
$r_1=r_0\frac{e^2}{e^2+8\pi\epsilon_0\hbar c}<r_0$
Therefore the electron moves closer to the proton rather than farther from it (an attractive force.)
A: Photons aren't like billiard balls. A photon isn't a particle in the sense of a hard, massive bit of stuff, it's a "chunkyness" of the Electric and Magnetic fields. When we say protons and electrons are attractive because they exchange photons, we mean protons create EM fields, and electrons create EM fields that cancel out those created by the proton.
The fields themselves are exactly the same as the classical EM fields, the only exception is that they can only be made and cancelled out in chunks. The reason that photon exchange can be attractive is precisely the same reason classical electric fields can be attractive.
A: For all the people looking to know more about particle interactions, you could go to check Matt O'Dowd from PBS Space Time's excellent video about virtual particles:
https://www.youtube.com/watch?v=ztFovwCaOik
