Main advantage is the mentioned above lower unsprung weight that increases the responsiveness of the suspension through reducing the inertia of the moving part of the suspension.

Second important advantage is the fact that the fork stanchions aren't in the path of dirt from the front wheel as is the case in a classic fork - this is the most important advantage of an upside down fork in my opinion as it increases the fork's life, reduces it's maintenance requirements and makes it very easy to completely shield the fork stanchions  . That's why it's a standard for cross motorcycles and starts getting into enduro bicycles.

As to other advantages mentioned in other responses - I'm wouldn't count on those. I'll explain:

1. Stiffness - upside down fork can be stiffer but only using much stronger tubing, this is obvious when you look at the image of a classical and upside down forks side by side.

On an upside down fork you have no connection between sliders (lowers), while on a classical setup you do have an Arch connecting those and making that part of the for very stiff. This makes the for very resistant to twisting stresses. Additionally the distance between the wheel axle and the sliders arch is te smallest possible distance between those components, and that also adds stiffness.

In an upside down fork you have a huge distance between the fork crown and the wheel axle, and as mentioned before the sliders have no other connection between them except the wheel axle. So all the forces going through such a fork have a big leverage on the stanchions and those have to be very beefy to be comparable to a classical design. Additionally a beefy through axle is an absolute must for an upside down fork as without it the wheel would twist in relation to stanchions (this is because the sliders are only solidly connected via the wheel axle).

2. Weight - if you create an equivalent stiffness forks of both designs with similar cost and materials because of point 1 it's highly doubtful that the upside down fork would be stiffer. This assumption comes from the fact that upside down forks are always positioned as higher end forks and they use better materials and machining, and usually those are the factors that cause those forks to be lighter, not the fact that they're upside down design.

3. Brakes mounting - force exerted on the sliders and stanchions depends only on the size of the disk brake nad the friction force that the caliper can exert on the disc brake - the leverage of this force in relation to the wheel axle is always the same for the same brake setup and it doesn't matter how the caliper mount looks like. Also it's important to note that not a single type of brakes exist that can be used only on upside down fork and not on classical design.

Main advantage is the mentioned above lower unsprung weight that increases the responsiveness of the suspension through reducing the inertia of the moving part of the suspension.

Second important advantage is the fact that the fork stanchions aren't in the path of dirt from the front wheel as is the case in a classic fork - this is the most important advantage of an upside down fork in my opinion as it increases the fork's life, reduces its maintenance requirements and makes it very easy to completely shield the fork stanchions. That's why it's a standard for cross motorcycles and starts getting into enduro bicycles.

As to other advantages mentioned in other responses - I wouldn't count on those. I'll explain:

1. Stiffness - upside-down fork can be stiffer but only using much stronger tubing, this is obvious when you look at the image of a classical and upside-down forks side by side.

On an upside-down fork you have no connection between sliders (lowers), while on a classical setup you do have an arch connecting those and making that part of the fork very stiff. This makes the fork very resistant to twisting stresses. Additionally the distance between the wheel axle and the sliders arch is the smallest possible distance between those components, and that also adds stiffness.

In an upside-down fork you have a huge distance between the fork crown and the wheel axle, and as mentioned before the sliders have no other connection between them except the wheel axle. So all the forces going through such a fork have a big leverage on the stanchions and those have to be very beefy to be comparable to a classical design. Additionally a beefy through-axle is an absolute must for an upside-down fork as without it the wheel would twist in relation to stanchions (this is because the sliders are only solidly connected via the wheel axle).

2. Weight - if you create equivalent stiffness forks of both designs with similar cost and materials because of point 1, it's highly doubtful that the upside-down fork would be stiffer. This assumption comes from the fact that upside-down forks are always positioned as higher-end forks and they use better materials and machining, and usually those are the factors that cause those forks to be lighter, not the fact that they're upside-down design.

3. Brakes mounting - force exerted on the sliders and stanchions depends only on the size of the disk brake and the friction force that the caliper can exert on the disc brake - the leverage of this force in relation to the wheel axle is always the same for the same brake setup and it doesn't matter how the caliper mount looks like. Also it's important to note that not a single type of brakes exist that can be used only on upside-down fork and not on classical design.