High input resistance - FET
For a MOSFET the high input resistance is caused by the isolation layer between gate and channel (blue in the picture):



The layer is made of SiO\$_2\$, which has an extremely high resistance of 10\$^{16}\$ &Omega;\$\cdot\$m, and is one of the best insulators existing.

That means that the voltage applied to the gate has no way to go, so there won't be any current, apart from a small leakage current (not through the SiO\$_2\$). MOSFET input opamps may have input resistances as high as 10\$^{13}\$ &Omega;.