We study the quasiparticle excitation and quench dynamics of the one-dimensional transverse-field Ising model with power-law (1/r^{α}) interactions. We find that long-range interactions give rise to a confining potential, which couples pairs of domain walls (kinks) into bound quasiparticles, analogous to mesonic states in high-energy physics. We show that these quasiparticles have signatures in the dynamics of order parameters following a global quench, and the Fourier spectrum of these order parameters can be exploited as a direct probe of the masses of the confined quasiparticles. We introduce a two-kink model to qualitatively explain the phenomenon of long-range-interaction-induced confinement and to quantitatively predict the masses of the bound quasiparticles. Furthermore, we illustrate that these quasiparticle states can lead to slow thermalization of one-point observables for certain initial states. Our work is readily applicable to current trapped-ion experiments.