An equation involving variables such as
$$ \tag{1} (a+b)(c+d)=ac+ad+bc+bd $$
doesn't in itself mean anything until you choose values for each of the variables. Once all of the variables have values we can do the arithmetic to evaluate each side of the equation to a number and see if the numbers are the same. Then, and only then, does the equation become something that is either true or false.
There are some equations -- such as (1) above -- with the property that they end up being true no matter which values we choose for the variables. That's a useful property; we call those equations "identities" or (in some cases) "rules".
However, we also work with equations that may or may not end up being true when we choose values for the variables -- for example
$$ \tag{2} 5a+2a=14 $$
which becomes true if $a$ is $2$ and false if $a$ is $3$.
We can rewrite such equations using rules, to get for example
$$ \tag{3} 7a-14=0 $$
and the point of the rewriting is that the value choices for $a$ that makes (2) true are exactly the same as the value choices for $a$ that makes (3) true.
This property is what we mean when we say the rewriting is valid, and a rewriting is valid whenever we can prove (in whichever convincing way we can think up) that the value choices that makes one equation true are the same ones that make the other true.
So in the end it all comes down to calculations on actual numbers, and the symbolic algebra on letters is just a slick way to predict how those calculations will come out before we choose which numbers to do it with. It's useful and time-saving but not in itself mysterious.
(The above is true at the algebra-precalculus level. In higher algebra we meet examples where variables really are things in themselves and everything does not straightforwardly reduce to actual values, such as formal polynomials over finite fields. But one shouldn't try to wrap one's head around those cases without being completely comfortable about the less advanced setting).
