The given language is set of strings that end with either 00 or 11.
You can think of it as follows. State q1 remembers the fact that till now we have seen a 0, so when we see another 0 we move to state q2, which indicates that till now we have seen double zeroes. And if input ends here we should accept the string. Now if a 0 comes we will remain in the same state, since last two characters that we saw in input string are 0s. And if we get a 1, we go to state q3, which is the state representing single 1. Similarly, q4 is for remembering that we have seen two continuous 1s in the input.
Hope that helps :)
