"As far as I know, they used standard 8x57mm Mauser ammo in their Spandau machine guns, and I really don't see how an 8mm bullet could be made to explode, especially with the technology available a century ago."
Explosive bullets, as well as tracer and incindiary, in rifle calibers were developed in the lead up to World War I.
From "Flying Guns – World War 1: Development of Aircraft Guns, Ammunition and Installations 1914-32' by Emmanuel Gustin and Tony Williams" (I think Tony is a member here, he and I have chatted frequently about cartridges in the past, nice guy!)
The signatories to the Declaration of St Petersburg in 1868 renounced the use of any projectile weighing under 400 g which was either explosive or "charged with fulminatory inflammable matter." At that time, the appropriate calibre for 400 g shells was 37 mm, which accounts for the popularity of this calibre for decades thereafter. This limitation was generally adhered to until 1914. However, as we have seen, tracer, explosive and incendiary bullets were all developed and used even in rifle calibres during the First World War. After the fighting was all over, it was decided that the legal position should be clarified so in 1922 a Committee of International Jurists met at the Hague to consider a set of rules for aerial warfare. They decided that the use of tracer, incendiary or explosive projectiles by or against aircraft was not prohibited.
Explosive and incindiary bullets were used primarily against balloons. They were filled with hydrogen, so the best way to knock them down was to set them on fire.
It was quickly discovered that rifle caliber machine guns weren't very effective, so the French modified their M1914 Hotchkiss and the British their Vickers gun to fire the 11mm Gras cartridge, which the Allies adopted as their standard heavy machine gun round.
In one of his books Eddie Rickenbacker talks about using 11mm Gras rounds to bring down balloons, and having to use the cartridges VERY sparingly because they were in such short supply.
What I don't know for sure, though, is whether the Germans ever developed a similar machine gun for attacking Allied balloons.
They did develop a Spandau-type gun chambered in 13.2x92mmSR for use against Allied tanks, but it was introduced very late in the war and I'm not sure that any mver made it into service in the air.