Here's a non-technical answer:
Bonds are a form of debt. What the issuer is selling is essentially a promise to repay the principal (i.e. whatever price the buyer paid) and some interest to the buyer. (Note: since we're talking about debt, you could also think of the buyer as a lender.) In return, the issuer gets to use the money from the sale for whatever current projects needed funding in the first place.
Skirting some technical stuff, assume the yield is the same thing as the interest rate. When lenders think there's a higher risk of not getting their money back, they'll either 1) not lend in the first place, or 2) if they do lend, demand higher a higher interest rate to compensate them for the higher probability of not getting repaid.
When someone downgrades your credit, it's a statement about you: they believe you're at a higher risk of not fulfilling your promise to repay the lender. So any lender that does decide to take a risk and loan you money (i.e. buy your bond) is going to be in situation #2 above: they'll demand you pay more interest on the loan (here, higher bond yield) to compensate them for your riskiness. That makes borrowing more expensive for you since you now have to pay a higher interest rate than you would have if you had better credit.
Rollovers are a type of reinvestment.
At the heart of the matter, credit downgrades happen when lenders lose faith in your ability to fulfill your promises. This happens for a multitude of reasons.
As an example, S&P's 2011 downgrade of U.S. debt happened because they believed 1) that the pattern of running deficits most years might eventually put the U.S. at risk of not repaying (that is, the debt-to-GDP ratio might not be sustainable), and 2) that political leaders lacked a credible plan to tackle the problem.