There’s an aspect to the BFR that doesn’t get nearly enough attention, but to explain its importance requires a bit of backstory.
To efficiently travel to the outer planets requires a great deal of energy in the final stage of the rocket. Delta-V (the amount of velocity a rocket can impart to its payload) is the fundamental commodity of all rockets, and travel past low Earth orbit requires a great deal of delta-V on top of what you already spent to get to LEO. Because energy scales with the square of velocity, the energy contained in the final stage tells us a great deal about how good a rocket is for travel beyond LEO.
There are two main ways to achieve this. First is to use a propellant with a high energy density–namely, cryogenic hydrogen (LH2). Hydrogen is a wonderfully energetic propellant with the downsides of being awful to work with, expensive, and having a low volumetric density (it is 1/14 the density of water!). Despite these downsides, a number of rockets use hydrogen upper stages.
The other way is to use more stages. Effectively, this is putting kinetic energy into the upper stage. The second stage of a typical rocket isn’t going all that fast at separation, and has a fairly low kinetic energy–but the third stage has tremendous KE, as it is traveling at or near orbital velocity. This makes a significant difference in delta-V capability.
Some rockets do both–the Saturn V had three stages, with the upper two being hydrogen. This sent quite a lot of payload toward the moon.
Now, the BFR is only a two stage rocket, and it doesn’t use hydrogen. It uses methane, which is a little higher energy than the Falcon 9’s kerosene, but it’s not nearly as good as hydrogen.
So what makes it a fantastic interplanetary craft? The answer is orbital refueling. You can send the BFR spaceship into LEO, and then refill its propellant stores over five additional flights of a tanker craft.
What this means is that the BFR is really a three-stage rocket. The third stage is just the fully fuelled ship. Normally, the ship would be almost depleted of propellant once it got to LEO, and likely couldn’t even get to the Moon, let alone anywhere farther out. But propellant transfer solves that.
Not only that, but it is an absolutely enormous three-stage rocket. It is fully equivalent to a huge rocket with a ship as the third stage, and six full BFRs strapped together as the first and second stages. That is a 27,000 tonne rocket, and nearly 10 times as big as the Saturn V. Not only that, but it has a highly energetic third stage (since it is going at orbital velocity).
The beauty of course is that a rocket that large never has to be built. BFR is big, no doubt (already bigger than Saturn V), but it gains nearly an order of magnitude in effective “size” due to the propellant transfer. LEO is an excellent place to perform the transfer, being relatively easy to get to, but also far enough (in delta-V terms) from the outer planets to make the transfer worthwhile.
All of this is really aside from the fact that the BFR is fully reusable. Propellant transfer gives capabilities that are not practically achievable any other way–we are not yet ready to build 27,000 tonne rockets. Maybe someday, but even then, propellant transfer will enable even larger payloads.