How does vacuum sealing something in a mason jar help?

The idea is to remove oxygen and moisture and get a good seal on the jar. The vacuum pulls the lid against the jar, forcing the gasket more tightly against the rim. Note that the seal won't be as good as with actual canning, though, where the gasket is also softened by heat, so it molds better to the rim.

That may or may not actually help you out, though: not all food suffers terribly from oxygen and moisture exposure, and not all food safety issues are mitigated simply by vacuum seals.

Notably, vacuum sealing is not canning. (I've heard of people using vacuum sealers in that context, and that may be part of why your sealer came with that attachment, but it's not generally a safe usage.) What really matters here is what the food is, not what the storage vessel is. If it's something that'd benefit from vacuum storage in a bag, it'll also benefit in a jar. Everything still has to be stored the same way it would be without the vacuum seal.

The National Center for Home Food Preservation has a great article: Should I Vacuum Package Food at Home? The key point:

These machines may extend the storage time of refrigerated foods, dried foods and frozen foods. However, vacuum packaging is not a substitution for the heat processing of home canned foods.

Vacuum packaging is also not a substitution for the refrigerator or freezer storage of foods that would otherwise require it. In fact, vacuum packaging can add to the concerns associated with storing of these perishable foods (which are foods not stable at room temperature and requiring cold storage).

By far the biggest food safety concern here is botulism, because you're creating an anaerobic environment where C. botulinum will happily grow. Anything with any possibility of bacterial growth will need to be refrigerated or frozen to stay safe.

Probably the most common case where this is useful and clearly safe is non-perishable dry goods. That article gives nuts and crackers as examples. Removing the oxygen makes the oil in nuts go rancid more slowly, and removing the water vapor makes crackers go stale more slowly. Anything else with substantial oil content or crisp cooked starch (like chips) would benefit similarly. Mason jars might be a slightly better choice for some of these things since a flexible bag would be more prone to crushing them, but otherwise, the jar isn't the important part, the vacuum seal is.

To have less oxygen and / or other contanminates. Like is used with pumps on wine.

Not sure how many foods are negatively effected by oxygen.

Potato chips for one. Potatoe chips are also effected by moisture and this is less moisture.

According to this fats oxide to produce “off” odors and flavors.

Since nuts are sold in sealed bags / jars I suspect they are effected by oxygen. But that may be just for moisture. Found a reference that the oil in nuts will oxidize so low oxygen is good for storage.

According to a comment a vacuum also required to create a seal. It kind of makes sense to me but I am not 100% convinced a vacuum is required for a seal. For sure you are not going to hold a vacuum without a seal. A mod asked me add this information but I don't feel athoritative here.

With having products vacuum sealed, it removes at least a portion of the air out of the container, regardless as to if a vacuum seal bag or vacuum sealed jar. With vacuum sealed products, they last longer in the freezer, less (if any) freezer-burn, and helps preserve the flavor of the product being vacuum sealed. The difference between vacuum sealed and canning is the manner in which an item can be stored. As an example, spaghetti sauce (no meat)- when you vacuum seal the extra no meat spaghetti sauce, it still needs to be stored in the freezer. With the same product, if it has been put thru the canning process, it is able to be stored on a shelf, in a cupboard, in a pantry, with there's no harm done to it by it not being stored in the freezer.