Meaning / proof of these regex
For all three of your statements, the answer is that, generally speaking, the implications simply aren't as intuitive as, say, the associative laws. Look at the analogous problem with algebraic expressions: we have associative laws for addition and multiplication and we also have a distributive law that states that for any expressions $p,q,r$ we have 
$$
p\cdot(q+r)=(p\cdot q)+(p\cdot r)
$$
Eventually, these rules should be intuitively obvious. However, one implication of these rules, that 
$$
(p+q)^3=p^3+3p^2q+3pq^2+q^3
$$
is probably not as intuitively obvious as the rules used to derive that identity. The utility of the result above is that it can be used as a tool to simplify other more complicated problems.

It's the same for regular expressions: the fact that, say, $r(sr)^*=(rs)^*r$, is correct can be proven rigorously, but having done that you can use it as a tool to show that 
$$
aa+(aab)^*aa=aa+aa(baa)^*=aa(\epsilon+(baa)^*)
$$
should you ever need to. For example, there is a handy technique, involving what's known as Arden's lemma that can be used to produce a regular expression describing the language accepted by a finite automaton. Depending on how it's applied, this can produce several regular expressions from the same FA, so it might fall to you to show that the expressions are indeed equivalent, in which case the "implications" you listed might be handy.

The upshot (here's the tl;dr part), is that the implications you mentioned are simply tools you can use when needed: they've beed proven to hold, but there's no reason why they should be intuitively obvious.