A longer axle will allow the rings to clear the frame. 68mm will be the width of the bottom bracket shell, the metal cylinder at the base of the frame that the axle assembly fastens to. The axle assembly itself is known as the bottom bracket and its width is measured from one end of the axle to the other. Chainsets are generally designed to be compatible with a certain length of bottom bracket (BB). To use the chainset you've got, you'd need a longer BB axle. You've then got the issue of front derailleur compatibility, which Brucey discusses upthread. There. compatibility is determined by ring sizes and by the number of rings.
One way to check what length bottom bracket the chainset is made for is to get the model number of the chainset and make enquiries from the manufacturer (or look it up in a reliable database). If you get a chainset and you're not sure what BB axle length it needs, you can check through trial and error, fitting it to the bike and going from there. If the BB axle is too short, the chainset won't go on fully; if it's too long there'll be a big gap between the chainset and the frame and your pedals will be a long way away from each other (which might feel weird on long rides and could cause other problems).
If you then need to replace the BB for one that's compatible with the chainset, you'll need to remove the current BB using the correct tool and a long spanner. If the BB has been installed a long time it may take some work to remove it, as it could be tight. Then it may need lots of leverage and, if it's corroded, it may need treating with penetrating oil.
It's all in a day' s work for a decent bike shop. That's always an option.
----
Edit: Just realised you're using Hollowtech II, so the normal advice about bottom bracket axle length doesn't apply.