When should I use Konjunktiv I?

Konjunktiv I is used to indicate reported speech (when you are telling someone what somebody else has said).
However, Konjunktiv I is seen as very formal, and so it is rarely used in spoken German. You will mostly find it in academic and journalistic German, where an academic might be summarising someone else's argument, or a journalist might be retelling information from a source. In both situations, Konjunktiv I shows that someone else's words and opinions are being told, not those of the author. This also means that in an A-Level exam, where you will have to write a short piece of formal German, you should use the Konjunktiv I where possible: it is appropriate in that style of writing (and lets you show off your grammar knowledge!)
You should also be aware that many Konjunktiv I forms look indentical to Present Tense forms, something which happens with all verbs, except in the 3rd person singular (er/sie/es) and with 'sein'. If you are emphasising that a statement is not yours, but the Konjunktiv I form coincides with the Present Tense form, you should use Konjunktiv II instead to avoid any confusion.