Can and could, like the other modal verbs, have developed quite a range of meanings and uses. You ask how to explain the difference, and explanation can certainly help, but learning all the ins and outs of these verbs is a long process which requires plenty of experience, observation and experiment.
possibility vs. ability, and can vs. could.
1) Ability and possibility
Ability and possibility are similar ideas. If you’ve got the ability to do something, then it’s possible for you to do it – in principle at least, although there might be something that prevents you. And, conversely, if you haven’t got the ability to do something, then it isn’t possible for you to do it. Both can and could (and other modals, especially may and might) are used to express various kinds of possibility, ability, permission and potential.
2) Can and Could
Could, of course, functions as the past tense of can, and like other past forms, it sometimes simply indicates past time: In those days there was no security and anybody could walk in, day or night.
But, like the past forms of other verbs, it can also indicate things like tentativeness, indirectness, deference and a wish not to impose. Compare:
I wanted to have a word with you (now)I wondered if you needed any help (now)
I want to have a word with youI wonder if you need any help
The last two examples are more direct; the first two are more tentative, and possibly more polite – though that depends on other factors, too.
When we make suggestions, we often like to be rather tentative, so as to avoid giving the impression that we necessarily expect people to do as we say. Of course, it’s possible to make much stronger suggestions – e.g. I think you should listen to the radio – but we generally prefer to adopt a more ‘take it or leave it’ approach, and lessen the possibility of offence on either side. So that’s why You could ….. is so commonly used in suggestions (and, by the same token, Could you …..? is often used for requests). In fact, it’s so common that it makes sense to learn it as one of the standard formulae for giving advice and making suggestions (and to reserve You can ….., in similar contexts, for factual statements). And, by the way, it’s quite common to add always: Well, you could always listen to the radio. Of course, the well at the beginning isn’t necessary, but it also contributes to the general impression of tentativeness.