Cs133 and Se77 NMR measurements were conducted on a single crystal to study the local magnetic and electronic properties of the Fe-based compound CsFe2Se3 with a two-leg ladder structure. From the experimental results of the Cs133 and Se77 NMR spectra and the Cs133 nuclear spin-lattice relaxation rate, we found successive magnetic transitions from a paramagnetic phase to an incommensuratelike antiferromagnetic (AFM1) phase with a distributed internal field at TN1=176.5K and then from the AFM1 phase to another commensurate antiferromagnetic (AFM2) phase at TN2=148.5-152.0K. Internal fields of the Cs and Se sites obtained from the angular dependence of the NMR spectra are discussed in terms of the transferred hyperfine tensors governed by site symmetry. For the AFM2 phase, we show that the stripe-type spin structure occurs in two domains, which have magnetic wave vectors of Q=(1/2,1/2,0) and (1/2,-1/2,0). On the contrary, we observed a characteristic modulation of the internal field with magnetic fluctuation between the ladders in the AFM1 phase. To clarify why CsFe2Se3 has a large charge gap although it is a Fe2.5+ mixed valent compound, we investigated the hyperfine coupling constants and found that a hole is doped into the Fe site instead of the Se site. Based on this fact, molecular orbital and double exchange models are proposed for the electronic state.