Average variable cost is the cost that varies (changes) as production increases. As we increase production, marginal cost is the measurement of how variable costs change with each additional unit of production. If variable cost on average is decreasing, the marginal cost for each additional unit of production is less than the average variable cost, but as soon as the average variable cost starts increasing the marginal cost (the cost of the next production level) crosses over and becomes higher than average variable costs. Let me give an example (it is not letting me build a table):
Output Var Cost AVC Marginal Cost
10 $1000 $100
11 $1070 $ 97 $70
Notice Average Variable cost decreased & MC < AVC
12 $1130 $94 $60
Same thing as above
13 $1224 $94 $94
With increase in production AVC is flat and MC = AVC
14 $1324 $95 $100
With increased production, AVC increased & MC> AVC
Again since AVC measure total variable cost averaged over whatever is the level of production. So each level of production has a different average variable cost. Marginal Cost measure the amount that variable costs changes between levels of production. To say another way, if production increases by one unit, how much will variable cost change. So for marginal cost to be below average variable cost, average variable must be decreasing. For marginal costs to be above average variable costs, average variable costs must be increasing. See example above.
Now the next question is why does it cost average total cost at its lowest point. Total cost is the sum of average variable cost and average fixed costs. Therefore the only change in total costs is changes in variable costs, since fixed costs don't change. So the same logic as above occurs. The reason that total cost is changing is because the variable cost is changing. Marginal costs simplely measures the change in costs as a result of a one unit change in production. And variable cost is the only change. The logic above is still valid.