Detailed microscopic image of polymer crystallization from the melt is searched for by molecular dynamics simulation. In order to accelerate crystallization, a simple model of polymethylene chain is devised; the polymer chain is made of 100 beads (CH2 united atoms) connected by harmonic springs and the lowest energy conformation is a linear stretched sequence of the beads with a slight bending stiffness being imposed. A system of polymer melt, made of 80 chains of C-100, is placed between two parallel substrates that represent the growth surface of the lamellae. Initial melt at 600 K is rapidly cooled down to various crystallization temperatures, and the molecular process of crystallization is investigated. We first notice a marked layer structure in the melt near the substrate. We find that the layer structure leads to growing lamellae when cooled below the melting point. It is shown that the growing lamellae have a definite tapered shape and show thickening growth along the chain axis as well as normal growth. The molecular trajectory of the crystallizing chain at the growth front is demonstrated to be a sequence of processes of local adsorption to the growth front followed by stretching along the chain axis and the final fold into crystalline conformation. The chains are found to be very mobile showing active diffusion in the melt and also in the crystal.