Initially the cost of producing a successive unit of output is lower than that or previous unit. Or employing an additional unit of a factor of production results in production of more units of output than what was produced by the previous incremental unit of input. This is called increasing marginal productivity. This is also where the marginal cost of production is declining (or marginal product is increasing).

During this phase, the marginal cost is lower than the average cost of the previous units and that's why MC lies below AC and they both decline.

Then comes what we call, diminishing marginal productivity (or increasing marginal cost). Marginal cost starts to increase, but for some time it is still below AC, So during this phase MC is rising (but is below AC) and AC is still declining. MC then intersects the AC curve from below, and this is the point where both AC and MC are equal and AC is at its lowest point. Now, MC continues to rise and is already higher than AC, hence each successive unit of input leads to an increase in AC.

You can see that the AC curve takes a U shape because it declines first and then increases.