The title of the Nobel Lecture of the person who invented electrospray ionization was "Electrospray Wings for Molecular Elephants". By that he meant that this technique is good for very large molecules and it does not break them. The interesting part is nobody fully understands how the big molecules get a charge. There are two models and who can explain it better than John Fenn, the inventor. Have a look at his lecture: https://www.nobelprize.org/prizes/chemistry/2002/fenn/lecture/
Coming to the electron impact ionization, it is a physical contact (don't take it literally) of a very high energy electron beam with the molecule, the energy of this process is sufficient enough to knock out an electron and break other bonds too in the gas phase. This breakage is understood in terms of molecule's cross section. Note that in ESI, no free electrons are used to ionize molecules. The ionization process is a mix of solution phase chemistry (i.e., what ions were present, pH, etc) and the molecule's own properties. Not all can be ionized by ESI. Certain ions will suppress electrospray phenomenon like trifluoroacetic acid.