How to cut away portion of sub floor to inspect for leaks?
This is going to be complicated and expensive. 

First, you need to determine if the leak is coming from the exterior or the interior. I'd check plumbing fixtures above the area and see if you can trace it down. It could be running down a wall and into the space under your subfloor. 

Second, if it's coming in from the exterior, you need to know the extent of the damage. So, you will need to remove sections of the subfloor. (If you live on a slope, I'd start on the high side of the slope.) If you find a wet spot in one area, I'd move 8' -10' and remove another section. If it's wet you can assume it's wet between sections too. However, finding the point at which it's leaking into your basement is key. I'd assume the wetter the area, the more likely you're closer to the point of leak. 

If it's coming from the exterior, Possible leaking points: 1) joint between concrete wall and concrete slab, 2) joint or crack in floor, we call this hydrostatic infiltration, (it gets pushed up from the sub-soil), 3) backfill against the outside wall that is placed too high on the wall. (Water could be coming in over the top of the concrete wall.)

There are other possible places where it could be leaking, (I.e.: roof flashing, downspout, stormdrain, expansion joint, etc.,) but I'd start with  the 3 listed above. 

Repair will depend on where the leak is occurring, of course. I'd let your contractor guide you on that. 

Hmmm...so...when you bought the house was this leak disclosed to you?  If not you probably have a claim back to the seller. I'm suspicious of the "new carpet" in the basement because I don't think most people refurbish their basement unless they're trying to cover up a past problem in hopes of a sale. So, be sure to document what you find, take lots of pictures, and get a written statement from your contractor on what he found and if it looked old or new. I've testified as an expert witness for buyers who found leaks, settlement, etc. covered up. A good professional can tell if a leak is new or old...usually.