You're right, 300dpi means 300 dots (or pixels) per inch. It's also known as print resolution. Anything lower than that tends to look blurry or pixelated when it's printed.
Most monitors work at 72ppi (some are 90ppi), so it's hard to see the difference on screen. But this is how it works: say you have two images, both 6x9 inches. One is at print resolution (300ppi) and the other is at screen resolution (72ppi). On the computer, they'll look the same, but if you print, there will be a drastic difference. That's because the 300ppi version is 1800 x 2700 pixels, while the 72ppi version is only 432 x 648.
Even if you convert the 72ppi version to 300ppi, there are only so many pixels to work with. The image will print clearly now, but it will only be 1.44 x 2.16 inches big. It comes down to math (unfortunately!)
For this example:
1 inch = 300 pixels
Big version (started at 300ppi):
1800pixels / 300ppi = 6 inches
2700pixels / 300ppi = 9 inches
Small version (started at 72ppi):
432pixels / 300ppi = 1.44 inch
648pixels / 300ppi = 2.16 inch
It's best to start your files at 300ppi, because you can always size down from there if you need to. Whereas if you start in 72ppi and end up having to print at some point...you will basically have to start over.
I hope that helps a little!