The short answer is that average costs fall because fixed costs get averaged across more units.
In more detail:
Costs have two components: fixed costs (which do not vary with produced quantity) and variable costs (which do vary with produced quantity).
For example, if you produce newspapers your costs might include
- paying journalists to write and edit the content of the newspaper
- renting a building for the journalists to work in
- paying to print each copy of the newspaper.
Notice that you only have to pay the journalists and the rent once, no matter how many copies you print (so these are fixed costs). The cost of printing (paper and ink) will increase as you produce more copies of the paper because you will need more of these inputs, so these are variable costs.
Suppose it costs \$10,000 to pay a team of journalists to produce an issue of a newspaper (fixed cost), and \$1 to print a copy of that paper (variable cost).
If you sell one thousand copies then the total cost is $$\$10,000+(1,000 \times \$1)=\$11,000.$$
That means the average cost of producing each copy is $$\$11,000/1000=\$11.$$
Now suppose production increases to 2,000 units. Now the cost is $$\$10,000+(2,000 \times \$1)=\$12,000.$$ The average cost per unit is $$\$12,000/2000=\$6.$$
So output increased and the result is that the average cost per unit has fallen. This is a situation in which there are economies of scale. The reason this happens is that, as output increases, the fixed cost (here the \$10,000) gets spread over more units so that the average fixed cost per-unit decreases even if the variable cost does not. In particular, notice that in the first case the fixed cost gets averaged over 1000 units (a fixed cost of \$10 per unit), whereas in the second it is averaged over 2000 units (a fixed cost of \$5 per unit).