What is the mechanism for electrospray ionization?
How electrospray works has been hotly debated for decades.  Two competing theories have been put forward.  

Initial stages: theories agree on droplet collapse due to Rayleigh instability

The two theories agree on the initial stages of droplet formation.  An aerosol is formed by the extrusion of low ("nano", or 25 to 100 nL / min) to high (~1 mL / min) flows of eluent through a capillary charged to ~3000 V.  The volatility of the solvent used in electrospray techniques ensure that the aerosol droplets evaporate.  Charged spheres of liquid experience a self-repulsion due to electrostatic effects.  This repulsion is counterbalanced by surface tension.  As Lord Rayleigh noted in 1887, as long as $T &gt; \frac{Q^2}{16 \pi a_0^3}$, where $Q$ is the droplet charge, $T$ is the surface tension, and $a_0$ is the droplet radius, then the droplet is stable.  But as heat evaporates the droplet, $a_0$ shrinks, while $Q$ stays constant and $T$ nearly so.  Thus the droplet eventually becomes unstable, exploding into a mist of finer droplets, each of which has a smaller charge. What happens after that has been the subject of debate.

The two theories for what happens after droplet collapse


ION EVAPORATION.  This theory holds that as droplets shrink, eventually the field strength at the droplet surface is high enough to force remaining solvated ions into the gas phase, an effect called "field desorption".
CHARGE RESIDUE.  This model holds that charged analytes remain solvated in droplets through many cycles of droplet evaporation / fission.  Eventually, droplets contain on average only one analyte molecule.  Remaining solvent molecules evaporate, leaving the ion in the gas phase.


Current evidence suggests that most small molecules ionize primarily by ion evaporation, but that for large molecules such as proteins, charge residue mechanisms may be important.  In the ion evaporation mechanism, molecules or ions must translocate to the surface of a droplet before they can become gas phase ions.  This  means that molecules which preferentially segregate toward the surface of liquid droplets will ionize more easily.  Thus, the ion evaporation mechanism, but not the charge residue mechanism, predicts that surfactants will have higher ionization efficiencies than non-surface active agents.

An extension of the ion evaporation model


EQUILIBRIUM PARTITIONING MODEL.  This theory divides droplets into an electrically neutral core, where ion pairing maintains local electroneutrality, and a charged shell, from which small molecule ions can move into the gas phase by ion evaporation.  The amount of "excess" charge in the shell available for ionizing neutral molecules from the core is determined by electrospray flow parameters, in particular by the ratio of electrospray current to flow rate, or $[Q] = \frac{4.2}{F} \sqrt{\frac{K \gamma }{V_f \kappa_e}}$, where $F$ is Faraday's constant, $K$ is the conductivity of the medium, $\gamma$ is surface tension, $\kappa_e$ is the dielectric constant,  and $V_f$ is the volumetric flow rate. Conditions where excess charge is limiting could cause severe ion suppression.  For more discussion, see this paper.


Conclusion:

This answer is very long which should be a clue that this topic is complex.  Usually, saying that things are complex is a sign that people don't understand something very well, and that is the case here.  But in general, perhaps the closest to a consensus in the community seems to be that small molecules ionize primarily by ion evaporation, but that for large molecules such as proteins, charge residue mechanisms may be important.