When an explosion takes place, the contents get scattered in every direction. So how is it, that after the "Big Bang", everything only went in one direction, to form a flat Universe?
An explosion is an expansive movement in space.
The Big Bang is the creation and expansion of space itself. It expands at the same rate at every place in every direction (though the rate of expansion is changing with time).
When we speak of the shape of the universe (either flat, or positively or negatively curved) we mean curvature of this space itself to be such that the matter in the universe is just dense enough thus there is exactly enough matter so that expansion will stop expanding eventually at infinite time (flat spacetime), will expand forever (negatively curved spacetime) or collaps back to a singularity in a finite time (positively curved spacetime). Thus 'flat' or 'curved' does not mean that one of the spatial dimensions you (and we all) look at with our eyes is bent or not bent - it's rather bent or not bent in time.