I'm also a visual learner, so here is what I think should help you visualize it.

These are the three spaces --> _ _ _

_ _ _

R R R

O O O

Y Y Y

G G G

B B B

I I I

P P P

These are the seven choices for each space

So lets cut it down to 2 spaces to make it easier

Each choice of the first space has 7 choices for the next space

the same for each number going down the list

R has 7 choices for the next space, O has 7 choices for the next space, and so on

=> This makes 7 + 7 + 7 + 7 + 7 + 7 + 7 = 7 * 7 = 7^2

So if you repeat this method for 3 spaces, you get 7^3 which equal 343 ways.

Hope this helps! :)