Translation:The class has already started.
ている is not always the continuous tense in English though. It is for verbs that describe an action (i.e. eating, drinking etc) but for verbs that describe a change, the ている form implies that the change continues. So, since "to start" describes a change, はじまっています describes the continuation of having started = "has started". The same is true for example for verbs like dying (死ぬ), which also describes a change, where 死んでいる does not mean "is dying" but "has died". So in a way, your first answer is a literal translation. There is of course also context needed since some verbs can both describe a change and an action (smelting for example).