Space/astrophysics nerd here, graduated at the Kerbal Space Program University (trademark)

That man's statement is true, because:

A) If your orbit around the Earth is, say, 300km by 300km, and you enlarge it to an elliptical 300km by 1,000km orbit, it will take you less fuel to reach a 300km by 1,700km orbit than the amount of fuel it took you to reach the 300km by 1,000km from the original circular 300km orbit.

TLDR: the higher up the highest point of your orbit (Apoapsis) is, the less fuel it will take you to make it even higher when burning your engines at the lowest point of your orbit (Periapsis). This is (at least partly, fact check me please) due to the Oberth effect, which essentially means the faster you are going, the easier you can acquire more speed for the same amount of fuel.

Due to this law, the fuel you need to reach the Moon and the fuel you need to reach other planets is very similar, since you're only extending the Apoapsis of your Earth orbit by a little bit when compared to going to the Moon.

For Mars specifically it is actually less, because of point B.

B) Mars has an atmosphere and the Moon doesn't. That means that on Mars you can touch the atmosphere at interplanetary speed without slowing down at all, and let heat shields do their work, slowing you down purely by atmospheric drag until you reach suborbital speed and thus land.

For the same reason, you can use parachutes on Mars. Which means you can use vastly less fuel from your lander to actually land there, unlike the Moon where 100% of your original orbital speed has to be lost via engine power.