For example: London Control says:
BAW211 contact London Approach 118.1, call sign only.
What will the pilot say to approach on 118.1?
London Approach, BAW211.
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You always start a communication with the intended receiver, so in this case it would be London Approach, BAW211. What the controller is telling BAW211 is that London Approach has the relevant details of the flight so BAW doesn't need to communicate anything further or make a request of london approach. This helps keep the frequency clear of redundant communications.
As for why you always lead with the recipient in the communication, not only is it in the regs it just makes sense. Think of it this way, if you get on the radio and say 'It's Frank.' nobody will know who you're taking to, if you say 'Mary, it's Frank.' then it's clear who you are speaking to.
You always include the identifier of the addressee, so London Approach, Speedbird 211.
This triggers the air traffic control officer, who may be mentally busy with something else, to listen. If the London Approach bit is left out, there is an increased chance of the whole message being missed or misheard, leading to a need for additional communication which the call sign only method was designed to prevent.
According to the International Federation of Air Traffic Controllers' Associations, it does cause some confusion:
[...] However, as there is no internationally recognised version of this procedure some confusion is known to exist in the pilot community [...] (ifatca.wiki)
Your safest bet is always to check the local AIP; as an example they quote Netherlands AIP, and I have confirmed it:
EHAM AD 2.22 [...] 126.96.36.199 [...] While being transferred from Amsterdam Radar to Schiphol Approach, initial contact shall be restricted to SCHIPHOL APPROACH + CALL SIGN only in order to avoid channel congestion. In specific situations, Amsterdam Radar may request pilots to report additional information to Schiphol Approach in the initial contact. (lvnl.nl)