How "exactly" is Rabies transmitted?
This is actually far more complex a question than you probably know. The country you are writing from matters. There are some rabies virus variants that have a relatively low pathogenicity for humans (dog rabies is one of these; a relatively large innoculum is required, i.e. a bite, commonly a deep one with lots of virus-laden saliva.) The likelihood of canine-associated rabies developing from a non-bite exposure is extremely low; in fact, the exact risk is unknown.
In the US, enzootic (dog-to-dog) canine rabies virus has been virtually eliminated through vaccination and stray control programs, making wild animals (including feral cats) the primary concern. Depending on the country your friend lives in, dogs may not even be high on the list of worrisome sources.
Regarding canine-associated rabies,

Rabies control was accomplished [through dog vaccination programs, responsible pet ownership, and serious population control of stray dogs] in Europe, North America, Australia, Japan, Taiwan, Malaysia, and Singapore.

If you don't live in one of these countries/continents, dog-associated rabies is still a major concern. If you do live in the mentioned areas, you can probably dip your hand in dog saliva and not be concerned.
Rabies is usually contracted by a bite; that is most common.

The most dangerous and common route of rabies exposure is from the bite of a rabid mammal. An exposure to rabies also might occur when the virus, from saliva or other potentially infectious material (e.g., neural tissue), is introduced into fresh, open cuts in skin or onto mucous membranes (nonbite exposure). ...Exposures to bats deserve special assessment because bats can pose a greater risk for infecting humans under certain circumstances that might be considered inconsequential from a human perspective.
how exactly does the process of infection happen, and how can it happen?

The exact way it happens is not completely understood, but that answer won't help you convince your friend of anything. The virus is usually deposited in tissue near a cutaneous (peripheral) nerve. If it gains entry to the nerve, it makes its way to the brain with the typical disastrous results.
Having said all this, your friend is not going to stop worrying, because the virus is only the focus of an irrational fear. Though the foci of irrational fears are fairly stable, OCD is most likely the underlying problem, and conventional rational discourse doesn't really suffice in curing the obsession.
During 1990-2007, 34 bat-associated human cases of rabies were reported in the US: 6 cases reported a bat bite; 2 cases reported a probable bite; in 15 cases, physical contact was reported (e.g., the removal of a bat from the home or workplace or the presence of a bat in the room where the person had been sleeping), but no bite was documented; and in 11 cases, no bat encounter was reported, but the rabies virus was bat-specific.
Any encounter with a bat, even a dead one (as stated above), must be evaluated for possible post-exposure prophylaxis.
Unfortunately, in the US, animal rabies is common, and at least 23,000 persons/ year receive rabies postexposure prophylaxis (PEP). (It is probably considerably higher, as no reporting mandate exists.) With the elimination of canine rabies virus variants and enzootic transmission among dogs, human rabies is now rare in the United States, with an average of one or two cases occurring annually since 1960. 
In the US in 2013, of the three human cases reported, 2 were involved in organ transplants (raccoon rabies virus variant) and one was a Guatemalan (canine rabies virus variant).
In 2012, one human died from an exposure to a bat. He touched a bat under a bridge. He did not report a bite to a witness. He became ill while traveling, and died in Switzerland. A number of humans exposed to his saliva (including his Swiss caregivers) received PEP.
*Some people in high-rabies areas without the illness have been documented to have developed antibodies to rabies. The mechanism is unknown.
Human Rabies Prevention - United States, 2008
Pathogenesis of Rabies
A Single Amino Acid Change in Rabies Virus Glycoprotein Increases Virus Spread and Enhances Virus Pathogenicity
Human Rabies Prevention (Comment From a Canine‐Rabies‐Endemic Region)
Epidemiology of rabies post-exposure prophylaxis—United States of America, 2006–2008
Rabies surveillance in the United States during 2013
Use of a Reduced (4-Dose) Vaccine Schedule for Postexposure Prophylaxis to Prevent Human Rabies: Recommendations of the Advisory Committee on Immunization Practices