Particularly when referring to times, about can mean "nearly; almost". If I say, "I'll be there at about three o'clock," then I am not promising to arrive at exactly 3:00, but I will be there not too long before or after three.
So "right about now" softens the "right now" from meaning "at this instant in time" into something a little more relaxed. It feels more like "very soon" and is not as urgent as "right now." It also has a fairly informal register because it mixes an intensifying word (right) with a word that softens the intensity (about), which is a bit logically contradictory. You probably would not see this phrase in formal writing.
That said, right about now is a fairly common set phrase and a lot of English speakers will use it without thinking very hard about whether it means anything different from now. It's more likely to signal informality than anything else.
One other way you'll sometimes hear this phrase in spoken English is with a big pause in it:
Often, the vowel in "about" will be drawn out as well. This is used when the speaker knows something is going to happen in just a few seconds. They say, "right abouuuuuut..." to signify the "very soon," and then shout, "now!" when the thing actually happens.