In GR a massive particle falls along geodesics of curved spacetime because there is no “force” on it and it has to follow straight lines. But how can the weight felt by an object on the surface of earth be explained by this concept?
Whwn you stand on the ground on Earth, you are basically in a freefall on a geodesic, but the ground is in your way. What you feel is the ground keeping you from freefalling.
It is a misconception that the effects of gravity are caused by mass. This is due to stress-energy. Even a photon, that has no rest mass, does have stress-energy and has effects of gravity.
It is a misconception too that only massive objects will follow the geodesic. GR specifically explains this effect of gravity with the phrase "gravity bends the fabric of spacetime" because all known particles in the SM will have a altered trajectory when they interact with the gravitational field. Yes, even a photon's trajectory will be altered by the gravitational field of the Earth, and the photon will try to follow the geodesic. The reason you do not only see photons fall towards the center of Earth, exactly on the geodesic, is that the Earth has relatively low stress-energy, and it can only alter the photons trajectory a little bit.
Although you would think that as you stand on the ground, you are in standstill relative to Earth, you are still moving in spacetime, in the time dimension. The universe is set up so and the four vector is set up so that its magnitude needs to be c always. If you are in the gravitational field of the Earth, time dilation will cause you to slow down in the time dimension (time will pass slower for you relatively), but because the four vector's magnitude needs to be c, the spatial dimensions will need to compensate. You will accelerate in the spatial dimensions. You will accelerate towards the center along a geodesic. But the ground is in your way, so you feel the ground pushing up(or you feel yourself pushing down).
But how can the weight felt by an object on the surface of earth be explained by this concept?
I like to say that the reason you are stuck on the surface of the Earth is because the shortest distance between today and tomorrow is through the center of the Earth.
"Weight" is the sensation of that natural motion being constantly interrupted by the Earth, which is moving in a different direction between today and tomorrow.
As Feynman put it, all the interesting physics is where your feet hit the floor.
Classical physics says weight is a type of force you experience. But it technically is not a force by definition under relativity, since gravity from first principles is a pseudo force by definition, and it is not usually custom, to quantize these fields. (Yet this hasn't stopped gravitons being speculated).