So those are Enkei LS5 rims. The only size I can find in 5x112 (which is our bolt pattern) are 18 X 8 (18" diameter, 8" wide) and they have a 50mm offset which is not close to our stock offset and is the reason you need the spacer (ours is 35mm in the front, 22mm in the back)... basically by adding the spacer to the wheel that has 50mm offset, you push it out farther so its closer to stock: a 15mm spacer plus the 50mm offest of the rim would get you to the 35mm offset needed for the front...

The bigger issue in all of this, though, is that those wheels are marked as being bolt centric, not hub centric... I'm not familiar enough to tell you if they can be made to work but I'm guessing that's why you're being told you need new wheels...

*edit: I assumed these were 18" wheels but if they're 20" then they would probably be the 20x8.5 with 40mm offset