In aluminum, how does electricity travel through the surface oxide layer?

The native oxide that coats aluminum is slightly porous, and the pores tend to trap tiny amounts of moisture in them. This renders them electrochemically active and ever so slightly conductive. (In fact, for the aluminum oxide layer to grow in thickness in hot environments requires that both the aluminum atoms be capable of diffusing up through the existing oxide to reach oxygen in the atmosphere and the oxygen atoms be capable of diffusion down through the oxide to reach unreacted aluminum underneath the oxide.)

In order to render an oxidized aluminum surface pore-free, the aluminum piece must be baked in an oven with an oxygen atmosphere in it, to close off those pores.

In the absence of porosity in the oxide, the conduction mechanism is Frenkel-Poole emission, where a random thermal fluctuation will occasionally promote a bound electron into the conduction band, where it can then drift under the influence of an external field.

The engineering answer is that current doesn't pass through the oxide layer very well at all, and if you want to make a good contact to an aluminum object you have to be very careful about how you do it.

If you just press a copper wire (for example) against an aluminum one, you will get a very high resistance contact. Probably not 4 gigohms, but maybe on the order of hundreds or thousands of ohms, so there may be something to the earlier answers which suggest the oxide layer is fragile enough to break away and allow some contact.

But, for example, you can abrade the oxide away with sandpaper, and then make an air-tight connection of (or solder or weld) the other object to the aluminum before the oxide has time to re-form.

For small enough geometries (like the bond-wires used to connect integrated circuit chips to their lead-frames) you can pressure weld aluminum directly to other materials like gold or silver. This tends to deform the aluminum wire substantially, which must spread the oxide out enough to prevent it interfering with the contact.

Or you can use a chemical "coating" or surface treatment on the aluminum to keep the oxide from forming. One of these treatments goes by several names such as "Alodine", "chromate conversion" or "chem film". (Note: traditional Alodine treatment is not usable for products to be sold in Europe due the RoHS directive, but newer chemical treatments are available that are acceptable under RoHS)

Or you can use a very aggressive flux to displace the oxide during soldering. But this flux must be cleaned very thoroughly to avoid continued corrosion of your parts.

I believe @Maxim Umansky is correct in his comment: the breakdown potential of the aluminum oxide layer is just a few volts (see, e.g., Fig.8 in (J. Electrochem. Soc., Solid-State science and technology, October 1976, p. 1479). For a layer thickness of about 4 nm we get a breakdown voltage of a few volts.