What does "high-capacity magazine clip" mean?

Here magazine and clip refer to parts of a gun that contain bullets. They are not used in the same sense as a clip cut out from a magazine for reading.

High-capacity refers to the ability of the magazine to store many bullets.

A magazine is the (often detachable) part of a firearm that stores the ammunition. A clip is a slang (and technically incorrect) term for a removable magazine. The phrase "magazine clip" is redundant, rather like referring to a bicycle as a "bicycle scooter".

In this context, high-capacity means "holding more ammunition than some people think is acceptable", not "holding more ammunition than normal", since the magazines this term is used to refer to are the normal size for many firearms.

I love the word "magazine" because of its origin (From Online Etymology Dictionary):

1580s, "place for storing goods, especially military ammunition," from M.Fr. magasin "warehouse, depot, store," from It. magazzino, from Arabic makhazin, pl. of makhzan "storehouse" (cf. Sp. almacén "warehouse, magazine"), from khazana "to store up." The original sense is almost obsolete; meaning "periodical journal" dates from the publication of the first one, "Gentleman's Magazine," in 1731, from earlier use of the word for a printed list of military stores and information, or in a figurative sense, from the publication being a "storehouse" of information.

The intended use in the quote is a container for bullets attached to a gun, which carries a greater than usual number of bullets.

The magazine clip holds ammunition for a gun.

Some guns use "magazines," which hold cartridges for the gun and, with the aid of a spring, feed them into a chamber. A "high capacity" magazine holds many cartridges; the number that is considered "high capacity" is fixed by law in some places.

A "clip" is something completely different. It is used to store cartridges in such a way that they can be rapidly fed into magazines. There are many pictures on the internet that explain the difference between a magazine and a clip. There is a good picture here: http://miaanstine.com/2012/12/18/clip-vs-magazine/.

People who are not familiar with guns sometimes say "clip" when they mean "magazine." Accordingly, they call for regulation of "high capacity clips" when what they really want is regulation of magazines. Newspapers make this mistake on a regular basis, and the White House has done it also.

A "magazine clip" is a newspaper's awkward attempt to refer to gun magazines while continuing to misuse the word "clip." There is no such thing as a "magazine clip."