You rub the garlic in the pot to gain a subtle garlic flavour. Same idea as rubbing a sliced garlic clove on grilled bread, or on meat prior to cooking.
Raw garlic is also moderately antibacterial in nature, so I suppose that could be a reason too.
Assuming this is a meat fondue (AKA fondue bourguignonne), using oil instead of cheese:
There is no best or correct oil to use - each type of oil has its own characteristic flavour. However, a meat fondue generally involves heating the oil to 350-400° F (175-200° C), so you'll want to treat this more or less like deep-frying and use an oil with a relatively high smoke point.
That means, no unrefined olive oil, sesame oil or walnut oil, and be extra careful if you want to use butter (clarify it first). Canola, sunflower, peanut, grapeseed, etc. are all good choices, or any of the refined versions of olive/sesame/walnut with higher smoke points.
Personally I'd either use refined olive oil or a flavour-neutral oil like peanut or sunflower. Most of the time you don't want to be imparting any "foreign" flavours to meat, although I suppose you could go with refined sesame oil for a moderate Asian twist.
Best Answer
Any white with a sufficiently high acid content. The canonical fondue wine is Fendant, which is made in the valais region out of chasselas grapes, so any chasselas (see Wikipedia for a long list of alternative names) will work well.
One notable alternative is a dry champagne. This will make your fondue very light and fluffy, due to the carbonation.