Hoping for clarification of this very basic explanation of the abc conjecture An article online says:

The abc conjecture refers to numerical expressions of the type $a + b = c$. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities $a$, $b$ and $c$. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers-- those that cannot be further factored out into smaller whole numbers: for example, $15 = 3 \times 5$ or $84 = 2 \times 2 \times 3 \times 7$. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.

My question related to this sentence therein:

In principle, the prime factors of $a$ and $b$ have no connection to those of their sum, $c$.

I don't understand that statement, since $$c = (\text{the product of the prime factors of }a) + (\text{the product of the prime factors of }b).$$
That certainly seems like a connection!
I do not have strong math chops, and would appreciate an answer in English rather than in notation, to the extent possible. I fully realize I'm just not grasping what's being said, I'm certainly not critiquing the conjecture.
 A: The way I understand it, the meaning of the statement

In principle, the prime factors of $a$ and $b$ have no connection to those of their sum, $c$

is not that there does not exist any relation at all between them (we start by the assumption that $a+b=c$ as you have already mentioned in your post) but that there is no (yet known) deterministic relation between these prime factors. In other words, there is no (well not yet) statement, proposition, theory, algorithm that can give a full description of the prime factors of $c$, given the prime factors of $a$ and $b$.
P.S.: Apart from Mochizuki's recent developments of course.
A: You seem to assume that $a$ equals the product of its prime factors (and likewise for $b$), but that doesn't take into account multiplicity of factors. For example, the only prime factors of 72 are 2 and 3, but 72 is not $2\times 3$.  
Even if you could get $c$ from just the prime factors of $a$ and $b$, that wouldn't tell you much about the prime factors of $c$ (and especially about their sizes) unless you went through the work of factoring $c$.
