If you look in the dictionary the meaning of accrue you will understand what is accrued interest. Accrue means accumulation or addition and accrued interest means accumulation of interest, this concept is of particular significance when one is buying or selling the bond or fixed income security.
If you are holding your bond till maturity than it is not an issue because you will be the owner of the bond till last date and therefore you will receive the interest, the problem comes when you sell your bond in between the last interest rate payment and future payment date. It can be better understood with the help of an example where one can see how the calculation is done for accrued ROI. Suppose you hold $1000 bond on which ROI is 10 percent per annum and you receive interest every 6 months which comes to $50 every 6 months, now if you sell your bond after 3 months than the buyer will pay you the price of bond and also three months interest which will be $25 ($50*3/6) because for three months you were the owner and therefore the interest was accrued for 3 months.
Above example was made with simple figures, the calculation for accrued interest can be much more complex and tricky in real life as there people buy and sell bonds on daily basis which makes the calculation quite complex.