To be involved is to be rolled up inside something, tangled beyond easy separation. The word comes from Latin involvere — in- ('in') and volvere ('to roll') — meaning 'to wrap up, to envelop, to entangle'.
The image is textile: thread wound tight around a spindle, impossible to pull free without unravelling everything. When you become involved in a project or a relationship, you are etymologically rolling yourself into it, binding yourself to its turns.
Latin volvere generated one of the largest word families in English. Revolve: to roll again, to turn in circles. Evolve: to roll out, to unfold — Darwin's evolution is the unrolling of life. Volume: originally a roll of papyrus, the standard form of ancient books. Vault
The word waltz may also trace to this PIE root *wel- ('to turn'), through Germanic rather than Latin — a dance of turning.
Involved gained its modern participatory sense by the 17th century. But the tangling meaning persists: 'it's complicated' and 'I'm involved' carry the same implication of being wrapped up in something not easily escaped.