Why is 3,6-dimethylpiperazine-2,5-dione optically active in spite of having a centre of symmetry?
There is no evidence (e.g. aromaticity) to support your claim that the compound is planar. In fact, crystalographic studies [1] revealed existence of both cis- and trans-3,6-dimethylpiperazine-2,5-dione. Corresponding CCDC numbers are LCDMPP01 and TRDMPP01, respectively — feel free to open the links and play around with the 3D structures in JSmol to understand the difference.

According to Sletten, in cis (ʟʟ) modification the ring is puckered resembling a skewed boat conformation, whereas the ring in trans (ᴅʟ) molecule is nearly planar:


Figure 1a. cis-3,6-dimethylpiperazine-2,5-dione (drawn from LCDMPP01)


Figure 1b. trans-3,6-dimethylpiperazine-2,5-dione (drawn from TRDMPP01)

References


Sletten, E. Conformation of Cyclic Dipeptides. The Crystal and Molecular Structures of Cyclo-D-Alanyl-L-Alanyl and Cyclo-L-Alanyl-L-Alanyl (3,6-Dimethylpiperazine-2,5-Dione). J. Am. Chem. Soc. 1970, 92 (1), 172–177. DOI: 10.1021/ja00704a028. (Open Access)