Why do military turbofan engines use a low bypass ratio?

Because the priorities for military aircraft (engines) are different. While it is true that the high bypass turbofans have better fuel economy (in cruise) and are less noisy, the low bypass engines offer significant advantages when we take into account their intended use in combat aircraft, such as:

• The response of the low bypass turbofans to throttle adjustments is faster compared to the high bypass turbofans; the inertia is less and less air mass is involved (for increasing the velocity)- This is important during combat, when thrust requirements change rapidly.
• They have less frontal area, reducing the drag produced. For aircraft expected to fly at supersonic speeds, however briefly, this is important.
• Better thrust to weight ratio- 6:1 in Trent 1000 Vs 9:1 F119 (used in F-22 Raptor)- Even if the actual thrust produced by the low bypass turbofans is lesser, they produce more thrust per kg of engine, which means that the engine can be more compact in size.
• The low bypass turbofans are more efficient at higher speeds compared to the high bypass turbofans.
• The lesser size of the low-bypass turbofans mean that the aircraft can be made stealthier by 'burying' the engines in the fuselage, which is all but impossible in case of high bypass turbofans.

Finally It's not military vs civilian, but subsonic vs supersonic-capable

Note that subsonic military aircraft use the same engines as civilian aircraft, even if their names might be different.

• The KC-135 used initially the J-57 which was called JT-3C when used in the Boeing 707-120. Now they fly the CFM-56, which is used on the Boeing 737 and the A320.
• The C-5 Galaxy uses the GE TF39 which became the CF6 when mounted to a Boeing 747-100 or a DC-10.
• The Fairchild A-10 uses the GE TF34 which is called CF34 when mounted to civilian aircraft like the Bombardier Challenger.

No, the differences arise only when the aircraft is designed to fly supersonic. This requires a very different approach to the integration of the engine:

• Supersonic aircraft engines are mounted close to the centerline. If possible, they are straight behind the intakes, so the intake flow does not need to change direction. Exceptions like the SR-71 are rare.
• Supersonic intakes are longer and have sharp edges as opposed to the short, blunt intakes of subsonic aircraft. Also, most have a variable geometry to adapt to the very different flow conditions at supersonic speed.
• Since it is the job of an intake to slow down the air going into the engine, supersonic intakes cannot have a big capture area, or their spill drag in supersonic flight would be excessive. Supersonic engines need to create their thrust with much less airmass than purely subsonic engines. Forget stealth, this is the real reason for the smaller diameters of supersonic-capable engines.
• The nozzle of a supersonic aircraft is also variable, in contrast to the fixed nozzle of subsonic aircraft. This again helps to adjust it to the flow conditions, but in this case the major difference is between reheat on and off. Afterburning engines are capable of much higher exit speeds to compensate for their smaller diameter. They accelerate less air to a higher speed to create comparable thrust.
• The last point mentioned it, but it deserves a bullet of its own: Supersonic engines use afterburners in order to have enough thrust for going supersonic at all. The hot exhaust gasses have a much bigger volume than the cold intake flow which needs to be accommodated by widening the nozzle.

Note that the civilian Concorde used also a variable intake and nozzle and afterburners. It had an engine which was used on the BAC TSR-2 before, a supersonic military aircraft.

The real distinction is not between civilian and military, but between purely subsonic and supersonic-capable. Initially, both was achieved with the same engines. The J-57 mentioned above was also used on the supersonic F-100 military jet. Only in the 1960s did those lines diverge, and the subsonic aircraft grew ever bigger low-pressure compressor stages. These were again driven by the high-pressure cores which were used on supersonic aircraft.