I just read the text of the speech here. The key part that relates to your questions appears to be this:

Returning to the moon is an important step for our space program. Establishing an extended human presence on the moon could vastly reduce the cost of further space exploration, making possible ever more ambitious missions.

Lifting heavy spacecraft and fuel out of the Earth's gravity is expensive.

Spacecraft assembled and provisioned on the moon could escape its far-lower gravity using far less energy and thus far less cost.

Also the moon is home to abundant resources. Its soil contains raw materials that might be harvested and processed into rocket fuel or breathable air.

I think the key word is "provisioned". This is based on several "ifs" that remain to be seen if they're true. We could provision spacecraft on the moon IF ice or hydrogen is found there. IF it can be easily extracted.

The idea, it seems, is to land a nearly empty spacecraft near one of the lunar poles and fill the propellant, oxygen, and hydrogen tanks from lunar materials, then fly to Mars. This idea has a significant drawback, IMO, in that not only do you have to escape the lunar gravity well, you aren't in a good orbit for flying to Mars because you took off from one of the poles. You could probably get around this by doing a slingshot around the Earth to pick up extra velocity and to line up with the plane of the Mars ecliptic. The velocity you pick up from this slingshot maneuver might be enough to make up for the propellant it took to land on and leave the moon.

Personally, I like the idea of boosting the ice (if it exists) away from the moon to orbit where it would be processed to LH/LOX. One idea I've seen for doing this would be to use a nuclear powered steam rocket to boost the ice. However you do it, you want to avoid having to launch all of the propellant from the Earth's gravity well if possible. It takes a lot of money and energy to orbit something from the Earth's gravity well.