What are these  characters, and how can I use them?
They are mathematical alphanumeric characters, which look like glyph variants of basic Latin letters but have been encoded as separate characters, due to their special use in mathematical notations. Italic, bold face, and even use of a sans-serif form vs. serif form may carry an essential difference of meaning in mathematics. For example, a bold italic “a” may denote a vector, in a context when a normal-weight italic “a” denotes a scalar variable. Normally, such distinctions are made with styling or with markup, but the mathematical alphanumeric characters let you make the distinction in plain text, when desired or needed.

The shapes of these character vary by font, even though the basic idea allows less glyph variation than for normal letters. A mathematical italic letter can still take different shapes. So no, they do not look the same to everyone else. See e.g. some samples of mathematical italic a in different fonts.

Moreover, not everyone sees them at all. Few fonts contain them, and it is quite possible that someone is using a computer where no font has them.

So it’s a matter of characters, not fonts. And these characters “are intended for use only in mathematical or technical notation, and not in nontechnical text” (Unicode Standard, chapter 15, page 481).

They are not used much, but people might be using them without knowing what happens. If you use a sufficiently new version of Microsoft Word and enter a formula, using the formula mode, and type an “a”, Word will actually convert the character to mathematical italic a.

Normally, you cannot type these characters directly. You would need to use a character picker like the one mentioned, or some input method based on the Unicode number of a character. But it is possible to create a keyboard driver that lets you type these characters using normal keyboard keys and some special keys – or to programmatically convert normal characters to these characters, as Word does.

There is nothing Apple-specific about this. Input methods vary by system and software, of course, like for other characters.