The .slice() method of arrays returns a shallow copy of a portion of an array. It's written as a method on the Array prototype, so the array it operates on is the array in whose context it's invoked. Normally that'd be something like this:
var newArray = oldArray.slice(2, 4);
When you call the .slice() method with .call(), you can control the value of this explicitly. In your friend's code, he's passing a jQuery object as the this value. Because there are no other parameters, .slice() returns a shallow copy of the jQuery object in its entirety, as a new array. (Like some other similar methods, .slice() will work on anything that looks like an array: basically, anything that has a .length property and numerically-indexed properties of interest. A jQuery object fits that description.)
However, since jQuery already has a built-in method to return a plain array of all matched elements, your friend should not bother doing that anymore:
var plainArray = $('[id^=name]').get();
That does exactly the same thing. Using .slice() isn't wrong, but it's kind-of pointless.