I think I understand the idea of thinking about gravity not as a force pulling an object towards another object but instead a warping of space so that an object moving in a straight line ends up following a path that brings it closer to the object, like two people at the equator both heading North and ending up at the same point even though all they did was move forward.
What I'm not following is why the speed that the object is traveling would affect the path it takes if all it is doing is moving forward and it is in fact spacetime that is bending around the planet. I can easily understand this in classical mechanics as two forces counteracting each other, but I can't visualise what is happening in a model of gravity as warped space.
Imagine a large planet and two objects passing by the planet both on the same course.
One is slower than the other. The slow object gets captured by the planet and falls into an orbit (or to the planet itself if it is too slow to make an orbit). If I understand correctly this object is simply moving forward in space but space itself bends around so that its path now takes it towards the planet. But nothing has pulled the object off its original course.
The other, a fast moving object, has its path bent slightly but flies past the planet and off into space. Same thing, it simply moves forward and again its path is bent by virtue of space itself being bent
If these two objects are both simply moving in a straight line through the same bent space time, both going only "forward" how would the speed of one object cause a path that is less bend towards the planet than the other. Surely one just travels through the same equally bent space time faster than the other.
I'm sure I'm missing something, but can't find a good explanation, most explanations I can find online about viewing gravity as curved spacetime ignore completely the speed at which the object caught by gravity is traveling.
Follow Up
Just want to say thank you to everyone who answered this question, blown away by how much people were prepared to put into formulating answers. I've not picked an acceptable answer since I don't feel qualified to know which is the best explanation, but they are all really good and have all really helped expand my understanding of this topic.
Best Answer
You're using the wording "curved spacetime", but you're still only thinking "curved space" with an independent, linear time.
In your curvature model, you're assuming that moving through some 3D spatial point in one spatial 3D direction will experience the same 3D path curvation independent on speed (as if you'd shoot a ball through a curved tube). You'd certainly agree that a different initial 3D direction will result in a different path.
Now we are in 4D, meaning that two different initial speeds are two different 4D directions, and as time cannot be treated as an independent component, but is curved together with space, this easily results in a different path.