The twig-like things are the lower yard slings. I question whether they ought to be white; they're made out of rope. I think the white paint (or some of it, at any rate) was applied at a relatively late stage in the model's construction. Parts of it may well have been repainted at one time or another.
Anyway, the yard slings are fairly simple. If you take a look at each of the tops in the Revell kit you'll see (if I remember correctly) a small, oval-shaped hole just ahead of the big hole for the heel of the topmast. The sling was, in essence, a loop of heavy rope that was secured to the center of the yard, passed through that hole, and went around the back of the lower masthead (there was some sort of cleat to keep it from sliding down). James Lees's The Masting and Rigging of Engish Ships of War, 1625-1860 shows it as starting out with an eyesplice on the starboard side just above the top. The rope then leads down through the aforementioned hole in the top and has a thimble seized into it. The rope then passes up through the hole in the top on the port side, around the back of the lower masthead, and through the eyesplice in its own end, to be seized back on itself. A second thimble is strapped to the center of the yard; the two thimbles are connected to each other by a lanyard. The function of the slings is to hold the yard more-or-less permanently in its position in the vertical plane. (If for some reason the yard has to be lowered, the lanyard between the two thimbles is removed and the yard is lowered by means of the jeers. If my memory is correct - as it may well not be - the Revell rigging instructions include the jeers.)
The blob on each of the stays is called a mouse. (Maybe somebody thought it made the stay look like a snake that had swallowed a mouse.) It's a rather fancy piece of ropework made of lighter line than the stay itself; its purpose is to keep the stay collar at its proper dimensions, while allowing the stay to flex a little as the ship works in a seaway. Making a mouse to 1/96 scale would be pretty tricky, but there are plenty of ways to fake it. The easiest probably is simply to wind fine thread around the stay repeatedly until the necessary shape is built up, then fix the mouse into position with white glue. A little black paint after the glue's dry will make it look pretty much like the real thing.
If you're looking for a good introduction to nautical terminology, here's a good place to start:
http://www.all-model.com
On the home page you'll see links to several books, the full contents of which are on the web. Start with George Campbell's Neophyte Shipmodeler's Jackstay, which is a classic from the early sixties. It was originally published by Model Shipways as a guide to building that company's solid-hull wood kits, so some of the specific techniques in it aren't relevant to plastic models. But the basic information in it regarding terminology and rigging is valid for ship modelers of all sorts. Anybody who learned everything in that little book would be well on the way toward becoming a knowledgeable ship modeler.
Also available via the All-Model site is Wolfram zu Mondfelt's Historic Ship Models. That's a current favorite among lots of modelers. It contains considerably more detail about some topics than the Campbell book, and has lots of detailed drawings. I have only two real reservations about it. First - because it covers such a vast amount of material it doesn't have time to discuss any one subject in much depth. Second - it's extremely European in its focus. A lot of references to woods, materials, and tools will leave modelers on the other side of the Atlantic a little bewildered.
One other book I strongly recommend for people just getting into the hobby is John Harland's Seamanship in the Age of Sail. It's a wonderful, beautifully illustrated treatise on how the actual ships worked.
Hope that helps a little. Good luck.