The framework the Universe is built from (space-time) is 'flexible' and is affected (to a degree) by the *contents* of the Universe.
Particles with mass *deform* spacetime. The effect is very, very weak - but it's there, and it's cumulative.
You and I have hardly any effect, but a big ball of stuff like the earth deforms space-time quite a lot. It results in 'curvature' of space-time.
This is the reason why when you throw a ball across the park, it moves in an arc. The ball actually moves in a straight line, but the universe (space-time) is curved towards the mass of the earth, and the ball's path traces a curve.
If you find this hard to believe:
Think about the path the ball would follow if you threw it in outer space, where local space-time isn't dominated by a powerful gravitational field like the one the earth creates. The ball would move in a straight line!
Did you throw it differently? Or is the curvature of spacetime different?
If you fired yourself from a cannon at the same speed and direction as the ball, if you look at the ball while you both fly through the air, *the ball appears to move in a straight line*.
From your perspective, space-time 'curves' less when you move in concert with the ball, compared to if you were stationary (relative to the ball).