I was reading something in which the author said: "the longer the maturity, the more and more gaussian the gamma of your option is". What exactly is the author trying to say in non-math terms and what does it mean to become more and more gaussian?
This statement is both unclear and somewhat incorrect.
To be more understandable the author should have said "as the option maturity becomes longer, the curve of Gamma vs stock price takes on more and more a broad, symmetrical bell shape". So he is using the fancy word Gaussian as a synonym for Bell Shaped, not as a precise math term.
For short term options he could have said "on the other hand when maturity is short the curve of Gamma vs S looks like a sharp Spike or an Upside Down Icicle".
However, if you look at this graph, for example link you will see that the gamma curve is not truly symmetric and differs visibly from the Gaussian curve shown in math textbooks link2. So the statement is imprecise at best.
It's equivalent to saying the Gamma, in your case, becomes more normally distributed / resembles a normal distribution, which comes with a set of well known properties, such as being symmetrical and having a mean of $\mu$ and a standard deviation of $\sigma$.
Gamma, being the first derivative, indicates the rate of change of the options Delta. The Gamma becoming more and more Gaussian is a result of the Delta moving closer to a step function*. This is typically a sign of lower uncertainty, due to less time until maturity and/or lower volatility.
* A function that is zero when $S \leq X$ and then "jumps" up to one for $S>X$ for a call option at maturity.