First of all, don't apologize for asking a "basic" question. We all have to start somewhere and it's important to have a strong foundation in the basics before diving into more advanced topics.
To answer your question, let's first define what we mean by "representations of SO(D-1)". In the context of superstrings, we are considering a string theory in a D-dimensional spacetime. This D-dimensional spacetime has rotational symmetry, which is described by the group SO(D-1). A representation of this group is a mathematical object that describes how vectors and tensors transform under rotations in D dimensions.
Now, let's look at the states with m^2 = 1/α'. These states are created by the operators α^-2 and α^-1α^-1 acting on the ground state |0,k>. These operators are related to the creation and annihilation operators for the string oscillators, which describe the excitations of the string. In particular, α^-2 creates a state with two units of string oscillation energy, and α^-1α^-1 creates a state with one unit of string oscillation energy in each of two different directions. These states correspond to higher energy and higher momentum excitations of the string.
Now, why are these states considered representations of SO(D-1)? This has to do with how they transform under rotations in D dimensions. The state created by α^-2 can be thought of as a vector, because it has two indices (one for each oscillator) and transforms like a vector under rotations. Similarly, the state created by α^-1α^-1 can be thought of as a tensor, because it has two indices and transforms like a tensor under rotations. Therefore, these states form a complete representation of SO(D-1), as they include both vector and tensor states.
I hope this helps to clarify the solution to exercise 1.5 in Polchinski's book. Remember, it's always important to have a solid understanding of the basics before moving on to more advanced topics. Keep studying and asking questions, and you'll continue to build your knowledge and understanding of superstrings.
