The long and short of it was that they were short of choices yet had a hankering for a payback.

As the Wikipedia article states, the Finnish Legion was composed of Finnish Reds that had managed to flee Finland in the aftermath of the Finnish Civil War (officially January 27 - May 15 1918). The Civil War ended in a total defeat for the Reds and led to a period known as "White terror," where anyone involved in the Red movement or even suspected of "Redness" was in danger of being imprisoned in lethally makeshift prison camps and/or summarily executed. Those Reds, their family members and sympathisers who could flee did so, and a large group ended up in Murmansk area.

These refugees had scores to settle with the victorious Finnish White government, which, at the moment, was a de facto ally of the Germans: a division of German troops had landed in Finland in the last phase of the Civil War and participated in the final battles. The Whites were in general quite pro-German and during the spring and summer of 1918, Finland seemed to be on the verge of electing a German prince as king of Finland. (Prince of Hessen, Friedrich Karl, was actually elected in October, but the German defeat in the war resulted to a quick about-face and the "king" never even visited Finland.)

Furthermore, many Whites had their eye on Russian Karelia, which many believed ought to be united with Finland as Karelians were close relatives of the Finns - the main differences were religion (mostly Orthodox versus mostly Lutheran for Finns) and that the Karelians had lived under Russian rule for centuries, whereas Finland proper had been first part of Sweden and then a semi-autonomous Grand Duchy of the Russian Czar and not part of Russia proper - at least in legalese.

In this situation, with the World War still going strong, the Allies only correctly assessed the situation as strategically problematic: German allies, with possible German reinforcements, were about to carve a swathe of sort-of-still Allied territory. On the other hand, Finnish refugees with a bone to pick were available and some even asked for arms and help against the White government, so why not send weapons and supplies where they could be easily shipped, and let the Finns bleed? For the refugees, it wasn't as if there were that many opportunities for having something to eat, and given their hatred for the Whites, it wasn't exactly difficult to persuade them to sign up.

For the British, arming the former Reds had another advantage in that it avoided having to declare war to Finland. The British expedition to Murmansk and Finnish Petsamo had come close to Finns declaring war to Great Britain, so using Finns against Finns lessened the diplomatic complications.

As the Wikipedia article states, the Finnish Legion was then employed against White Finns only. In the end, they did not fight the "official" Finnish government forces, but participated in the fighting in Russian Karelia where Finnish volunteers of the Viena Expedition attempted to unite Russian Karelia with Finland by force of arms. In this campaign, White Finn volunteers and their Karelian allies skirmished variously against the Finnish Legion, Russian Bolshevik forces, Red Karelians, independence-minded Karelians supported by the British, and even British troops. The expedition, or the "kinship wars," were a failure and became somewhat of an embarrassment and foreign policy liability to Helsinki government, which withdrew what tepid support it had offered.

As for the British expedition to Murmansk, the defeat of Germany changed its priorities from preventing German or German-aligned Finnish encroachment to helping the Russian Whites in the Russian Civil War. As noted, the Finnish Legion had quite firmly established that it would not fight Bolshevik forces, but they were quite content to work against White Finns, so that meant that more British and White troops were available to use against the Bolsheviks.

Interestingly, while the Finnish Legion was repelling White volunteers in Russian Karelia, the British government had reached an understanding with the Helsinki government to use Finnish islands in the Gulf of Finland as a staging area for naval raids into St. Petersburg area - not that the fledgling Finnish government was in any position to prevent that though. These raids would have repercussions in 1939, as they were one of the justifications Stalin cited for securing a broader defensive perimeter for Leningrad.

If this all sounds a bit complicated, you are not alone - the settlement of the World War up north was often a three-way fight between various factions that had their reasons to work together sometimes while even fighting each other at other times.