First off, make sure your blade is sharp and doesn't have any nicks in it. That can cause some tearing and make it difficult to get through more than one layer without skipping and ripping some of the fabric. Also, make sure that you lay the cutter right up against the ruler at a 45 degree angle up, and a slight angle out, and press down firmly. I like to put my hand on the ruler kinda like a "spider" with my fingers up and bent, rather than down flat. I think it gives me better control over the ruler. They can slip around a bit if you're not careful, and that too will cause you to have a jagged, uneven cut resulting in some of those "strings" you mentioned. If you're cutting a fairly long piece, then just "walk" your fingers up the ruler as you cut. So, cut a little, move up a little, cut a little, move up...
Just take your time with this until it becomes second nature to place the ruler, your hand, and the cutter all in the right position.