Starts with narrow mouth, tongue does not touch palate (roof of mouth), instead starts tucked back in throat and then mouth spreads open.
Tongue is rolled, producing a somewhat buzzing/vibrating sound.
Wide part of the upper tongue touches the palate, and it curves down as the consonant is pronounced in full.
So the Japanese R is rather like a clipped "L."
Form an "L" but only let the tip of your tongue touch your palate and say it quickly. That's the best way I can think to describe it. It's like a combo "R" and "L" and can sometimes sound like a muffled "D" since it's more rhythmic.
And here's a handy webpage with examples!