Why do the Boeing 787 wings curve when flying?

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The wings of the Boeing 787 are so flexible because its carbon fiber material can be stretched more, and the high aspect ratio of 11 will magnify this effect. In flight, all you will feel is less shaking due to gusts, because the wing will dampen load changes more effectively. On the ground, the wings might have less tip clearance, because less in-built dihedral is needed – the rest is supplied by the wing’s elasticity in flight.
The influence on performance is slightly negative, but this is a very weak effect. It can be compared to the rolling resistance of a stiff bicycle versus one with a spring-loaded frame.

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