Despite spray cooling in the form of droplet boiling on a textured surface being a very promising phase-change heat dissipating method, the understanding of droplet/bubble two-phase dynamics in the nucleate boiling is extraordinarily limited. In this study, we report sessile droplet boiling on micropillar array surface in the nucleate boiling regime using a three-dimensional lattice Boltzmann model comprehensively. Effects of micro-pillar size on bubble behaviors inside droplet are discussed in detail, covering bubble nucleation, growth, coalescence, and rupture. For the micro-pillars with large side length or small spacing, nucleation sites are activated around micro-pillar top surface. The preferential activation location of nucleation sites is determined by temperature, confined space and fluid flow. In bubble growth stage, the variation of bubble radius with time follows the square root law, being consistent with previous experiments. Bubbles merge into a large central bubble beneath droplet for the short micro-pillars while into a vapor layer for the long micro-pillars. Emergence of large central bubble prolongs droplet lifetime but deteriorates heat transfer. In addition, increasing micro-pillar side length or decreasing micro-pillar height can delay activation of nucleation sites.