The eagle must hold on tightly to the rock because the cliff is so steep. The word "clasp" could make you think of a hair clasp or the clasp that holds a stack of papers together. The eagle doesn't have to struggle to stay upright.
he stands (line 3)
The entire first stanza builds to this anti-climax. Compared to words like "clasps," "ring'd," and "azure," the final word "stands" isn't very exciting. Think of someone standing watch or standing guard.
The wrinkled sea beneath him crawls (line 4)
We see a contrast between old ("wrinkled") and young ("crawls") in this line. Everything in the world below is either too old or too young. But the eagle, in Goldilocks terms, is just right. The eagle is a creature in the prime of its strength and vitality.
He watches from his mountain walls (line 5)
The eagle continues to "stand watch" over the landscape. The eagle's vision is a kind of skill – it can see things from much farther away than can humans. Still, the poem keeps the eagle's coolest skills in reserve, in order to build suspense.
And like a thunderbolt he falls (line 6)
Like a dive-bombing fighter plane, the eagle careens off the crag, and he relies on gravity to build incredible speed. His flight is pure energy. The verb "falls" pairs with "stands" at the end of the first stanza. "Standing" and "falling" are opposite actions.