Radioactive emissions cause ionisation when electrons are chipped off atoms, rather like bullets chipping bits off stonework. The electrons are negatively charged, so if we could sweep them up with a + voltage and measure the current they generate, we get an idea of how many electrons were chipped off by the radioactivity in a particular time. This page with animation tells you all you need to know. Here’s a picture.
The radiation passes through a thin window, ionises the inert gas inside, producing an avalanche of electrons which are attracted to the central positive anode. The electron current goes through a resistor and the voltage across it is a measure of the count rate. A calibrated counter records the count rate.