My understanding was that Zeppelins would fly at an altitude where they'd be out of reach from conventional planes, and yet there were apparantly multiple occasions where Zeppelins were shot down by aircraft. How was this done?
First of all, let's address the service ceiling (maximum altitude) which P-class Zeppelins and British planes could reach. -(British) B.E.2c – RAF 1a engine: 3,050 m (10,000 ft) -(British) Sopwith Pup: 5,600 m (17,500 ft) -(British) Sopwith Camel Biplane: 6,400 m (20,997 ft) -(German) Military Zeppelin P Class: 3,500 m (11,600 ft) Evidently, British planes were more than capable of reaching higher altitudes than the German Zeppelins.
How did they do it? In 1915, Reginald Alexander John Warneford of the British No. 1 Squadron, was able to successfully destroy an enemy Zeppelin aboard a Morane-Saulnier L fighter by flying over the airship and bombing it.
In 1916, William Leefe Robinson, a British pilot, shot down a German airship while flying a B.E.2c and using .303 Incendiary ammunition which ignited the flammable hydrogen gas inside the Zeppelin.
Here's a list of some of the weapons used against airships: -Ranken Darts: Explosive flechettes designed to be dropped from aircraft. -Buckingham Bullet (.303 Incendiary): Phosphorous-filled incendiary ammunition fired from Lewis and Vickers guns from fighter aircraft. -QF 6-pounder Hotchkiss Mk 1: 57mm naval gun. Four of these guns were installed on both HMS E4 and HMS E6 British submarines for anti-airship duty.