How do Costs relate to Diminishing Marginal Productivity? 

Adding additional factors of production (such as workers) generally leads to higher productivity. At some point, adding additional workers begins to add less and less productivity.

Each factor of input has an associate cost. The total costs of production will begin to rise more rapidly as output increases. If the increase productivity does not outpace the increase in costs, then you have diminishing marginal productivity.

Diminishing marginal productivity occurs because, with fixed inputs, each additional unit of input  contributes less to overall production.