The breakdown between how much of your payment is going toward principal and interest is very important.
The principal balance remaining on your loan is the payoff amount. Once the principal is paid off, your loan is finished. Each month, some of your payment goes to pay off the principal, and some goes to pay interest (profit for the bank).
Using your example image, let's say that you've just taken out a $300k mortgage at 5% interest for 30 years. You can click here to see the amortization schedule on that loan. The monthly payment is $1610.46.
On your first payment, only $360 went to pay off your principal. The rest ($1250) went to interest. That money is lost. If you were to pay off your $300k mortgage after making one payment, it would cost you $299,640, even though you had just made a payment of $1250. Interest accrues on the principal balance, so as time goes on and more of the principal has been paid, the interest payment is less, meaning that more of your monthly payment can go toward the principal.
15 years into your 30-year mortgage, your monthly payment is paying $762 of your principal, and only $849 is going toward interest. Your principal balance at that time would be about $203k. Even though you are halfway done with your mortgage in terms of time, you've only paid off about a third of your house.
Toward the end of your mortgage, when your principal balance is very low, almost all of your payment goes toward principal. In the last year, only $513 of your payments goes toward interest for the whole year.
You can think of your monthly loan payment as a minimum payment. If you continue to make the regular monthly payments, your mortgage will be paid off in 30 years. However, if you pay more than that, your mortgage will be paid off much sooner. The extra that you pay above your regular monthly payment all goes toward principal.
Even if you have no plans to pay your mortgage ahead of schedule, there are other situations where the principal balance matters. The principal balance of your mortgage affects the amount of equity that you have in your home, which is important if you sell the house. If you decide to refinance your mortgage, the principal balance is the amount that will need to be paid off by the new loan to close the old loan.