You need to give the water a path that's more attractive (ie, lower) than your driveway. With your driveway being considerably lower than the road, it's currently the attractive path. If you only build up your driveway, you'll create a pond on the left side of the driveway since your lawn is also lower than the road.
If the ultimate place the water goes (ie, the low spot, drainage path) is off to the right of your driveway, you need to provide a path to that point. This can be a wide, shallow ditch (or swale) that you can run the lawn-mower in, (that is, such a wide shape that it's not something the mower gets stuck it, but something that can be mown) and drive through where it crosses your driveway, or it can be a deep, narrow ditch with a pipe in the bottom and crushed stone on top. You can also have a swale everywhere but the driveway and a culvert (pipe) where it crosses the driveway.
It does not look like you can raise the driveway much without exceeding the height of the garage floor, so other than reshaping the driveway so that any water on it runs off to the right, you should be mostly digging down to get water moving away from the garage rather than building up.
If making a swale, it's well worth cutting the sod, rolling it back, digging the swale, and rolling the sod back on - the grass limits erosion, and sod (which is already there, if you just cut it) is a lot easier than getting seed started in a swale.