Mnemonic devices are tools to help you remember things ("mnemon" is Ancient Greek for "to remember"). They work by associating things you can't remember well to things you can easily remember, then prompting you the thing you can remember easily, which in return makes you remember the thing you're having a hard time to remember.
"0", for example, might remind you of an egg, because the shape of the symbol looks like an egg. Or, it remind you of the anime character Zero from Code Geass, because that's both the name of the character and of the symbol.
The list in the OP is a so-called peg list: A list of numbers, each associated with something the number reminds you of.
The Animu-and-Mangoer's Mnemonics:
0,
1,
2,
3,
4, Four-leaf clover (Yotsuba/Sup Forums)
5, Yami Yugi (KoG winning by getting 5 exodia pieces; post-#s ending in 5)
6,
7, a red line (SEVEN YEARS HANDDRAWN)
8, 8-man (Hachiman)
9, Cirno (9)
10,
11,
12, Mumei (TWELVE YEARS OLD), party van
13,
14,
15,
16,
17,
18,
19,
20,
21,
23,
24,
25, christmas cake
26,
27,
28,
29,
30, Wizard (because, you know...)