Newcomer,
Velocity is defined as speed AND direction, if the direction or speed or both changes then the velocity is changed by definition. Also acceleration is a change of velocity, i.e. if either the speed or direction changes (the velocity changes) then acceleration occurs by definition.
Also, the term Einstein created is "spacetime" - one word, rather than "space-time"
A couple of simplistic examples of Einsteinian Gravity I use:... You are walking on the sidewalk in your neighborhood and you see that the path ahead is blocked by bunch of garbage cans. So, you automatically change direction and go around the cans and proceed on in the direction you were going. Or, you are traveling on a highway and it starts to go uphill, so you change direction and go uphill following the upward curve. So, as a mass encounters a "warp" in spacetime it changes velocity (or accelerates) to follow the curvature caused by the warp.
Sources: I use my cell phone since it is an on-line encyclopedia (Wikipedia). Also, the "Learning Company" has excellent video courses on the subject (www.thegreatcourses.com). For a deeper, more rigorous discussion on the topic, search "Einstein's Field equations"