This form of Euclid's Lemma follows easily from basic laws of GCD arithmetic. First I will present the
proof using the standard notation $\rm\: (a,b)\:$ for $\rm\: gcd(a,b),\: $ immediately followed by a proof employing a more suggestive arithmetical notation, denoting $\rm\:\gcd(a,b)\:$ by $\rm\ a \dot+ b\:.\:$ Because the arithmetic of GCDs shares many of the same basic laws of the arithmetic of integers, the proof becomes more intuitive using a notation that highlights this common arithmetical structure.
The proof below uses only said basic GCD laws: the associative law, $ $ the commutative law, the distributive law $\rm\, (a,b)c = (ac,bc)\, $ and the GCD-specific law $\rm\: \color{#C00}{(c,1) = 1}.\: $
Lemma $\rm \ \ (ac,b) = ((a,b)c,b)\,\ [=\, (c,b)\ \ {\bf if}\ \ (a,b)=1]$
$\begin{align}\rm {\bf Proof}\ \ \ ((a,\,b)c,\ b) &\rm =\, (ac,\,bc,\,b) = (ac,\ b\color{#c00}{(c,\ 1)}) = (ac,b)\\[.3em]
\rm (a\dot+b)c\dot+b\ \:\! &\rm =\,\ ac\dot+bc\dot+b\, = \ \, ac\dot+b\color{#c00}{(c\dot+1)}\ =\ ac\dot+b \end{align}$
Notice how the suggestive notation in the second proof invites us to exploit our well-honed arithmetical intuition regarding the associative, commutative and distributive laws of integer arithmetic during analogous GCD arithmetic proofs. $ $ For a less trivial example see a similar proof of the Freshman's Dream $\rm\ (A,B)^n\, =\ (A^n,B^n)\ $ for GCDs and cancellable ideals.
The motivation behind this powerful abstract axiomatic approach becomes much clearer should you go on to study ideal theory and/or divisor theory (and you should, for there lies much beauty).
For further discussion and generalizations see this post and see my menagerie of posts on GCDs.