What do these transistors do?

Without making me feel like a total idiot can someone please explain to me what the function of these transistors are?

I'm working on a PIR circuit with a counter. I want to tie the output of the PIR to the input of the counter, the connection diagrams on both the datasheets call for a transistor on each (or when used together just one transistor I imagine?) but I need help understanding what the transistor is doing?

PS, I never could thoroughly wrap my brain around transistor theory in tech school.

The output of your PIR is an open collector transistor. Basically it acts as a switch that applies a ground to a load and completes a circuit.

Lets see if I can explain a little better. Refer to attached diagram. When the PIR is not detecting an object, transistor Q1 is turned off. The input to the counter is pulled to the +V2 through R7. When PIR detects an object, Q1 turns on acts like a switch so now bottom side of R7 is brought close to ground. What this does is create a pulse to the counter. The counter detects the pulse edge and increments. Does that make sense?

A very short transistor tutorial--

Transistors are three-terminal amplifiers, that is, they take a small input signal and make it larger (when properly biased). The output and input terminals share a common terminal (The emitter for a bipolar and the source for a FET)

Both bipolar and FET (MOSFETs are most common) transistors have a high output impedance in their active "on" region (when being used as an signal amplifier and not a switch) which makes them act rather like constant current devices. That is the current through them is relatively independent of their collector or drain voltage.

A bipolar transistor amplifies the current that goes into the base-emitter junction so that the collector-emitter current is many times the base current (as determined by the transistor Beta or Hfe gain value). A bipolar transistor has a relatively low input impedance, since the base-emitter junction looks like a diode, and you must limit the base current with a resistor to avoid damaging the device.

A FET transistor amplifies the voltage between the gate and source terminals into a current from the drain to source (as determined by the transistor gate threshold voltage and the transconductance). A MOSFET input looks like a capacitor and thus has a very high input impedance.

If a bipolar transistor has a large base current applied (typically 1/10 of the collector current) or a MOSFET transistor has a large gate-source voltage applied (typically 10V), then the transistor will act as a switch with a low on-resistance.

Does that help your brain to do a little more wrapping?