General rule of keeping speed in a car is the less you're turning the steering wheel, the more speed you can carry, and the better you can accelerate. Running over the inside kerb means you can turn in earlier but also use less steering input as you don't have to turn as sharply. You can therefore take the turn faster because, as described above in previous posts, you have effectively increased the radius of the curve (i.e. makes the curve less tight). This assumes running over the kerb doesn't upset the car's balance or break any bits, and is something for the racetrack. Sometimes, if you REALLY cut the kerb tightly, you can drop the inside wheel off the inside and into the grass, which is often slightly lower than the kerb. The wheel will then hook onto the kerb, using it a bit like a railway track, allowing you to carry more speed again. You can experience this basic principle on some corners on The Nurburgring in GT4, where the tarmac has a distinct depression on the inside of some bends, particularly around the first part of the complex. Drop a wheel into the depression there and you'll feel the car being helped around the corner. Mark Hales described using this to his advantage in some race or another. Unfortunately I can't remember where he was racing or what he was driving.
Don't try that at home.
Like Touring Mars, if it's a graded kerb, not a vertical one, or a low mini roundabout, I also like to just nudge it when driving past, just for practice in anticipating where exactly my car's wheels are. It's a challenge, like hitting a particular catseye.