How do we wrap into our head that the universe is flat when we can see stars around us like enclosed in ball?
Is the flatness geometric or cosmic? I don't know how to ask this question to be sincere :(
We are here only discussing geometry: two parallel lines stay parallel. This is what 'the universe is flat' means. Remember, a cylinder is flat. There are actually several acceptable topologies for which the universe is still flat.
So if two people at different part in the universe draw straight parallel line, will the lines meet or will they go on forever?
They will never cross. They can either go forever, or come back to their starting point. It depends whether the universe is finite or infinite.
Exactly, the "Flat" or Euclidean Universe doesn't mean two-dimensional either, because we obviously have depth in our universe. As @lemouth said we can theorize and measure this if we place two parallel lines in space; throughout all of space they should remain parallel.