What is the point in Wizard creating RtR Forests and M13 Forests, for example? It seems like it would be a lot more efficient and cost effective to everyone involved if these basic land types were not specific to a set. Wizard wouldn't have the cost of manufacturing (as many of) them and we wouldn't need to buy new ones just to stay legal.
You don't need to buy new ones just to stay legal. You can use any card that has been reprinted in a recent set in Standard. All the basic lands are printed in just about every set.
Secondly, some people like collecting different land arts, so they produce lands with new arts every set, and many people try to collect one or more of each.
As has been said in the comments, the point is largely the art; functionally the basic lands are always the same and always legal. Sets are designed with a coherent theme and world in mind, and the lands should fit with that as much as the other cards should.
You can see a fairly convincing demonstration of this in view of lands in Innistrad and Avacyn Restored. They're all the same locations, but while in Innistrad they were dark, cold, and barren, in Avacyn Restored they're brighter and warmer, often with more life. It's well worth looking at the link for all the full-sized art, but for example, here's one of the mountains:
As long as it has the same English name as a legal card, a white or black border and a normal back (including shape and size), you can use the card.
Keep in mind that a card's characteristics is determined by Gatherer, not by what's printed on the card. If you use a card that has received an errata, Gatherer overrides the characteristics printed on the card. For example, despite the print showing otherwise, a Mountain from Arabian Nights is a Basic Land — Mountain with no ability besides the one granted by rules.
- Some promotional cards have a Gold border. These aren't legal for play.
- Alpha's rounder corners rule them out, and oversized cards are novelties.
As others have noted, lands fall under the same rule as any other Magic card - any version sharing the same English name is legal in any format where any version is legal. (If they reprint Lightning Bolt, you can use your 20-year old Bolts). The same is true for the basic lands (because I don't think there's been a moment where the five basics haven't been legal. :) )
As for why they reprint them, there's a really simple reason - accessibility for new players. At any given moment, there is a new player starting the game, and no matter what product they pick up, Wizards wants to make sure there's lands for them.