How does a microelectrode work?
A microelectrode is quite literally a small electrode and they come in a variety of shapes.

The glass pipette electrode you are specifically referring to is mostly used for patch clamp experiments. Patch clamp experiments are performed using various configurations: 


Source: Leica

So basically there is the cell-attached configuration, where a patch of cell membrane is sucked slightly into the pipette. Upon stronger suction the membrane is ruptured and the whole-cell patch clamp configuration is obtained. Here, access is gained to the cell, but a high-impedance seal is maintained between pipette and cell membrane. Lastly, there is a configuration where the patch is broken off the cell to exclude effects of the entire cell. This patch configuration is often used to obtain single-channel recordings. Dependent on the exact technique an inside-out or outside-out patch configuration is reached (see the picture above).

Hence, the glass pipette is basically only there to obtain a proper seal (cell-attached), to gain access to the cell (whole-cell), or to break off a piece of membrane. Notably, in whole-cell mode the pipette provides a means to administer compounds into the cell (think ions or drugs).   

The recording electrode is situated inside the pipette and is a simple metal wire (platinum or platinum/iridium) and nothing different from any microelectrode as you rightfully state in your question. The electrode wire simply picks up a voltage difference (or current flow) between the interior of the cell (in whole-cell mode) and the exterior. The exterior is measured using a reference electrode outside the cell. 

Furthermore, current can be injected to stimulate the cell, or to current-clamp the cell at a specific current. Likewise, voltage can be applied to voltage-clamp the cell. Therefore, an inert material is often chosen such as platinum or platinum/iridium to maintain integrity and prevent electrolytic breakdown. Electrodes are made out of metal to provide a low-resistance means to pick up currents or voltage-differences.

Voltage differences or currents can be amplified with an amplifier, registered and analyzed. See the following picture for the basic recording setup: 


Source: Robinson et al. (2013)

Reference
 - Robinson et al. Front Neur Circuits 2013; 7: 1-7