We study the physical mechanisms that cause the offset between low-redshiftand high-redshift galaxies on the [OIII]/H$\beta$ versus [NII]/H$\alpha$``Baldwin, Phillips & Terlevich'' (BPT) diagram using a sample of localanalogues of high-redshift galaxies. These high-redshift analogue galaxies areselected from the Sloan Digital Sky Survey. Located in the same region on theBPT diagram as the ultra-violet selected galaxies at $z\sim2$, thesehigh-redshift analogue galaxies provide an ideal local benchmark to study theoffset between the local and high-redshift galaxies on the BPT diagram. Wecompare the nitrogen-to-oxygen ratio (N/O), the shape of the ionising radiationfield, and ionisation parameters between the high-redshift analogues and asample of local reference galaxies. The higher ionisation parameter in thehigh-redshift analogues is the dominant physical mechanism driving the BPToffset from low- to high-redshift, particularly at high {\nii/\ha}.Furthermore, the N/O ratio enhancement also plays a minor role to cause the BPToffset. However, the shape of the ionising radiation field is unlikely to causethe BPT offset because the high-redshift analogues have a similar hard ionisingradiation field as local reference galaxies. This hard radiation field cannotbe produced by the current standard stellar synthesis models. The stellarrotation and binarity may help solve the discrepancy.