The simple explanation is that many consumer-relevant interest rates are either explicitly tied to the LIBOR or implicitly tend to move in tandem with the LIBOR. For example, many adjustable-rate mortgages and student loans are pegged to the LIBOR, so when this rate goes up, so do the rates on loans whose payments represent a significant proportion of consumers' budgets.
So why has the LIBOR been in the news recently? Well, it turns out that the LIBOR is calculated not by looking at the supply and demand of actual interbank loans, but instead by asking large banks what they think the LIBOR is. This system isn't necessarily problematic, however, as long as the banks whose opinions determine the LIBOR don't have an incentive to either under or overstate the true rate of interest. Unfortunately, banks have a number of lines of business whose profits are very much affected by interest rates, so there are pretty strong incentives to manipulate these interest rate reports in order to benefit the banks, even when this happens at the expense of others.
This market manipulation hurts borrowers directly because they often end up paying higher interest rates on their loans than they would if interest rates were reported accurately. Interest-rate manipulation can also hurt the general public indirectly, since reporting artificially low interest rates can make a bank seem more financially healthy than it actually is and thus obscures relevant information for investors and customers.
The New York Times has a great graphic that sums all of this up in handy picture form, for those of you who are visually inclined. Any way you look at it, there is compelling evidence that banks do in fact engage in fixing the LIBOR in order to suit their own interests rather than to convey a proper price to the market.