Color Theory is a really vast area of expertise; I had entire classes on that subject alone when studying Creative arts and Graphic design.
To give you the short answer, there are 2 things you should learn in order to argue colors intelligently:
- The basics of color theory
- The psychological meanings of colors.
For the fundamentals of color theory, start with the color wheel. This is what color tools like coolors use to create color schemes, and this is what Adobe used when they introduced color schemes in their software.
Here is a basic one:
It shows the basic relation between complementary colors, which are 2 colors opposite on the color wheel.
There are many different possible relations between colors that make a harmonious color scheme. This one shows triads, which are 3 colors spaced evenly:
The other popular schemes are Analogous (adjacent colors) and Split complementary (a color, its complementary and the 2 colors analogous to that complementary color). You can purchase spinning color wheels in art supplies stores if you need to show clients.
You can find more details on the most classic schemes here.
Of course, you can stray from these principles and still have harmony. Sometimes, dis-harmony is what you're going for. It's similar to note progressions in music theory; you can come up with your own, but some have been established for valid reasons.
The other aspect to consider is the psychological meanings of colors, which is meanings and associations that we give colors, as well as the emotions that colors inspire. For example, red is hot and brings excitement and anger. Blue is cold and evoques calm. It might sound very holistic, but it is used a lot in advertising; colors are rarely chosen randomly. Crown Royal is purple and yellow because these colors are associated with royalty.
Here is a good summary of this:
Source (as well as more details for each colors)
So this is a very good starting point, you should definitely read more on the subject. As with any type of theory that has to do with creativity it's not set in stone. It can (and should) be argued, but those principles have been used for centuries by all the great artists.
On a final side note, while this will allow you to explain the logic behind color choices, I also recommend that you pick your battles. Sometimes, no amount of logic will make a client chance his mind if he wants weird colors that you think look ugly.