It is a colloquial use of bag as a verb, in widespread use in Britain in the mid-twentieth century. One still hears it today, but it has become slightly dated. I can well imagine its continued use in Australia.
The following examples from the OED illustrate that it has two senses, labelled as 6a and 6b. It is the second one which implies an idea of reserving something. It is based on the principle, often with children( but not always) that possession is nine points of the law - bags something and it is yours.
a. To seize, catch, take possession of, steal. To add to one's ‘bag’ (bag n. 9). fig.
1818 T. Moore Fudge Family in Paris vi, Who can help to bag a few, When Sidmouth wants a death or two.
1857 T. Hughes Tom Brown's School Days ii. iv. 304 The idea of being led up to the Doctor..for bagging fowls.
1861 F. Max Müller in Sat. Rev. 23 Feb. 197/1 A stray story may thus be bagged in the West-end of London.
1879 Bell's Life in London 28 June 4/2 Whom Mr. Hornby very smartly ‘bagged’ at mid-on.
1936 J. Dos Passos Big Money 72 He was almost bagged by a taxicab crossing the street.
1940 I. Halstead Wings of Victory i. ii. 59 Pilot Officer Elliott..has now bagged two.
1943 C. H. Ward-Jackson Piece of Cake 11 To bag, bagged, to hit by aerial gunfire; shot down.
b. To claim; reserve. Used esp. by children (see quot. 1914 and bags I phr.). colloq.
1914 Concise Oxf. Dict. Add. 1045/2 Bag, (also, in school slang) claim on the ground of being the first to claim (I b., but usu. bags I or bags, first innings!).
1923 J. Manchon Le Slang 56, I'm going to bag the best chair.
1948 R. A. Knox Mass in Slow Motion vii. 68 The other girl bagging the hot-water pipes first.
1968 Listener 29 Feb. 269, I bags be Anthony Wedgwood Benn.