I know how circuits work, I don't know where the electrical side is coming from.
Quite simply, a galvanic cell is a single-cell battery. There are two hard components to the circuit: the anode and the cathode. The electrolyte and the metal-to-metal connection are what create a complete circuit.
When two dissimilar metals are electrically connected (such as when they touch, or you connect them with a wire) and placed in the same conductive electrolyte (in our case salt water), the difference between the reactivity between the metals and the saltwater creates a voltage difference between the two metals. Since they are connected, electrons naturally flow from the low voltage source to the high voltage source (don't forget electrons are negative charge).
The low voltage side of the cell is the anode, and the high side is the cathode. At the anode, water and various contaminants react with the metal, oxidizing it. When they do so, electrons are liberated from the newly single hydrogen atoms, which become H+, or more simply, a proton with no electron. They remain in solution in the water, increasing the acidity. The oxygen atom from the water molecule reacts with the metal in the cathode, corroding it, and often releasing part of it in solution, depending upon the exact metal present.
Meanwhile, the cathode doesn't react with the surrounding fluid directly because it is receiving an excess amount of electrons, more than enough to prevent it from corroding. Instead, it serves as a sink for electrons, and actually dumps them into the solution. For example, it may convert the single H+ ions into hydrogen molecules (consisting of two hydrogen nuclei and two electrons) by giving the ions electrons. It may also convert water and dissolved oxygen into OH- (hydroxide) ions. (The balance of hydronium and hydroxide ions determines the acidity or alkalinity of the solution. At a pH of 7, the concentration of hydronium ions equals that of hydroxide ions, meaning pOH is also 7).
So this forms a complete circuit. Electrons are stripped from the solution at the anode as the anode corrodes, and are redeposited in the solution at the cathode, which does not corrode at all. Break the circuit either by separating the electrolyte or by cutting the wire (or other connection) between the two pieces of metal, and the flow of electrons stops, and the galvanic corrosion halts. (Note that other types of corrosion may still occur at much slower rates.)
This circuit is exactly what batteries are, including the lead-acid battery under the hood. By stacking a number of cells in series, you can increase the total voltage, and do useful work with it. Even on large structures with galvanic protection or active galvanic corrosion, the current flowing from cathode to anode (conventional current) is usually measurable with just a simple multimeter. Break the circuit at the connection and you can still measure the voltage potential between the two items.
To protect against it, there are a few options:
(1), you can simply electrically isolate the two metals. This is where rubber washers, sleeves and other components come into play. For example, the TJ isolates the body from the frame, probably to try and prevent accelerated corrosion due to somewhat dissimilar metals. (Of course the brake lines messed that up.)
(2), you can use a sacrificial metal to erode to protect your metal. By choosing a more anodic metal and placing it in direct electrical contact with the metal you want to protect, you can stop corrosion of the protected metal, as long as it is in the same continuous electrolyte as the sacrificial metal. This is what zinc blocks are for on boats, and why steel sheet and steel bolts are often galvanized or zinc coated. Even if you chip some plating off, the corrosion of the surrounding zinc prevents the steel from corroding. However, once the zinc runs out or is no longer in the same continuous electrolyte, protection ends. Hence why you rarely find zinc blocks on cars, but instead things are galvanized and zinc plated where possible.
(3), you can apply an active voltage driver to the circuit and keep the voltage too high for electrons to flow. This is used very often in underground and underwater pipelines, and in large coastal structures (such as oil rigs). Instead of letting current flow freely, you drive it in reverse. The better the electrical isolation is between the two metals, the less current you need. This is exactly the same principle as "float charging" a battery. Simply keep the applied voltage a bit higher than the anticipated electropotential difference, and no significant corrosion occurs.
In regards to #3, if you drive it with a high enough potential in a solution containing metal ions, you can deposit a different type of metal onto the surface of the base metal. This is called electroplating. This is how you can deposit a silver or gold layer onto a cheaper metal only a few atoms thick. In the reverse potential, it is even a method of galvanization called "electrogalvanization". Electrogalvanization is chemically equivalent to charging a battery. Galvanic corrosion is discharging that same battery.