As Helena has indicated, "y que" shows that "que muestra avances" is a second relative clause parallel to "que están [sic, for "está"] demandando la población", and the antecedent of "que" is therefore the same in both: "la inversión". In the second, however, it is the subject of "muestra". So it's saying that "la inversión [...] muestra avances".

This is poorly expressed, because "mostrar avances" should really mean "showing signs of progress", which would mean that the investment is manifestly increasing, and I don't think this is what they mean to say. I think they must mean that there are signs of progress as a result of the investment. So I suggest "producing signs of". I admit that I can't be certain of this, but I think in context it really has to be the intended meaning.

As for "para apalancar", I read this as an example of para + infinitive with passive sense. Just as "cosas para tener en cuenta" means "things to be borne in mind", so "avances para apalancar" means literally "advances to be leveraged", which I would prefer to express as "progress to be reinforced" (or "bolstered"), and I think it would be clearer if we said "progress that needs to be reinforced", which is the implicit meaning. In my suggested version, "need" is plural because I'm making it agree with "signs"; it sounds more natural to me like that.