Scrim is a fine, thin layer of fibers (a very thin fabric) that is used to needlepunch batting through. Needlepunching through scrim makes a batt very stable and even. It can make hand quilting more difficult.
I'm not familiar with the Connecting Threads batting. My favorite batting for machine quilting is Mountain Mist Blue Ribbon 100% cotton, a very traditional batting that creates the traditional crinkly look (and becomes softer with washings). I would never use this for hand quilting, though. Quilter's Dream Request cotton is a good batt for hand quilting. People who prefer a more modern look (no shrinkage in the batting) like polyester batts.