You can install an interior weeping system similar to what you would normally do on the outside, but without the waterproofing.
Start by jackhammering out about two feet of floor around the edges of your garage, as close to the walls as you can get. Dig down until you reach dirt, and if that's not at least two feet down, keep going. Now, put down a couple inches of 3/4" crushed gravel, and then install weeping tile (it's not tile anymore; it's 4-6" corrugated plastic drain pipe with a nylon mesh sock over it). Make sure the sock is continuous across any joins you make in the weeping tile for corners etc. This is a good time to direct those drain holes down into this trench, where the water they drain will flow into the weeping tile. You'll also need a sump, or a connection to a downhill storm sewer; direct the weeping tile to this sump pit or the storm drain. While you're doing all this a center drain can't hurt too much; you can tie it into the weeper wherever it's convenient, just make sure the garage floor and the drain flange are level, or that the garage floor slopes slightly to the drain.
Now, you can install waterproof sheathing on the interior walls contacting the concrete. Normally this product is designed to go outside the foundation walls, but in cases where that's infeasible it can work this way too. The idea is to trap moisture that weeps through the wall behind the sheathing, where it will then be directed down the wall and under the slab to the weeper. So, you'll install the sheathing, which should have an air space between wall and most of the layer, and make sure the end is down in the trench by the weeper.
Backfill the rest of the trench with 3/4" crushed gravel up to the slab footing, then patch the slab with new concrete. Your garage should stay much drier.
If you don't have gutters and the grading around your house isn't sloped away, I'd start there. It's relatively cheap to do and will eliminate most problems. It's also nice, if you have a door at the bottom of a roof edge, to not enter under a waterfall.
Once you've done that (or if it's not possible to fix the grading) and the foundation is still wicking water, then an exterior water proofing system and french drain makes sense. This involves a painted on sealer, a free draining membrane that's attached to the wall (this gives a direct path for any moisture that reaches the wall to run directly down to the drainage pipe), and a perforated drainage pipe running around the perimeter of the foundation that drains well away from the house. If a system like this isn't installed during initial construction, it's often left to a last resort because of the expense and labor of removing any landscaping and soil that is against the foundation.
Best Answer
It doesn't look to me like the slab has shifted, since the crack appears level and isn't terribly wide. I would contact a concrete contractor, one that specializes in repairs, if there is erosion under the slab they can usually pump a slurry under it to fill in the void. Is your garage floor "Floating"? Look around the edges, if the floor goes under the walls, it's part of the support of the building and you want to consult a professional immediately. If the floor butts up against the wall, it's a "floating" floor and isn't part of the structure, however you may still have erosion under the wall footing, and I would still consult a professional, however a floating floor will be easier for a contractor to break up, install proper drains and such so it won't erode underneath again, and re-pour.
You could get an endoscopic USB camera from Amazon for about $20 that you could probably snake under the slab and inspect it, but I would consult a professional. Most of these types of things are cheap to fix now, but could end up costing thousands to fix later.