Learn Italian: Gli presto il mio quaderno. - I'm lending them my notebook.

In Italian, short indirect object pronouns like gli (meaning 'to him' or 'to them') almost always come right before the conjugated verb. So, instead of saying 'I lend to them,' Italian structure works like 'To them I lend' (Gli presto).

I thought gli meant 'to him'. Why is it translated as 'them' here?

You are completely right that gli officially means 'to him'. However, in everyday spoken Italian, it is extremely common to use gli to mean 'to them' as well. The strict grammar-book word for 'to them' is loro, which goes after the verb, but you will hear gli used this way all the time in real life.

Why do we say il mio quaderno instead of just mio quaderno?

In Italian, you usually need a definite article (like il, la, i, le) before a possessive word. Since quaderno is a masculine singular noun, we use il mio (literally 'the my notebook'). The main exception where you drop the article is when talking about singular family members (like mio padre).

How do we know who is doing the lending if there is no subject pronoun like io (I)?