In a simplified view, there's a motor and a gearbox and when they both are connected the gearbox makes the motor work. You only can shift the gear when both motor and gearbox are disconnected.
For this part, there's the clutch. The clutch is the connection between motor and gearbox.
To engage means to fit two parts of an machine together and the opposite to disengage is then to free those parts.
From a technical point of view, when you engage the clutch, the motor and gearbox are connected and you are able to drive and when you disengage the clutch, the motor and gearbox are not connected.
Thus, if the clutch is engaged and motor and gearbox are connected, it's not possible to change the gear.
But I found this example in OALD:
Engage the clutch before selecting a gear.
Technically, this means you put motor and gearbox together and then change the gear. That's not possible as discussed above.
And this is taken from Merriam-Webster's Learners Dictionary:
I have to learn how to let the clutch out smoothly.
Actually, the part you need to do smoothly is to lessen the pressure onto the pedal and eventually to take the foot away. But by taking the foot away, you rather put the clutch between motor and gearbox; hence, technically you let it in and not out.
And in Collins Learner's Dictionary, they say:
Laura let out the clutch and pulled slowly away down the drive.
You can only drive when the clutch is between motor and gearbox, so actually you can't drive when the clutch is off.
In my mother-tongue we literally say to clutch out when we put our foot onto the pedal and to clutch in when removing our foot from the pedal. This corresponds to the technical function.
I wonder why in English the terms seem to be used opposite to what they actually represent. Why do you let the the clutch out when actually putting it in, and why you engage the clutch when actually take it out?