Marginal cost of debt -- why does it increase with more funding

So I understand the breakpoints and that costs of capital increase after certain breakpoints.

But I’m not quite understanding why… why does debt cost increase after X amount? Why does equity costs increase after Y amount?

I don’t think it necessary has to do with Cost of Capital increasing after a certain $ amount, but it has more to do with the leverage ratio and the balance sheet strength of the organization. 

Let’s take a practical view here.

Let’s say you are a lender to a company that currently has no debt and Free Cash Flow of $10m for the year. Would you be comfortable lending that company $10m at LIBOR + 1%? You probably would because they will likely be able to pay you back quickly. However, let’s say that same company already had $100m in debt. Would you be comfortable lending that same company $10m in additional funds at LIBOR + 1%? Probably not, as you would demand a higher premium because the company has a higher risk of default and may not be able to repay the loan. 

Likewise for shareholders, if a company is more leverage, the company has a higher risk and should be reflected in the volatility of the share price (ie. higher Beta) resulting in shareholders demanding a higher equity premium for investing, resulting in higher implied cost of equity. 

Agree with the above. More risk requires more reward, therefore higher CoD and higher CoE. Breakpoints seem to establish the points in which there is an uptick in required return based on the underlying capital structure.