The Spanish meterse (“to get involved with”) comes from the Latin mittere (“to let go.”) They sound like they might be opposites, but they’re broadly aligned: it’s all about going somewhere, figuratively. Getting involved with something is just getting to your destination!
From this Latin root, we get a whole slew of English words, such as:
Basically, all the ‑mit words–even the awesome, but usually forgotten, manumit!
What all of these words have in common is, going in a particular direction: the permission to go there; the acceptance to go there; the submission to see if you can go there; and even the opposite, just not going there at all!
Note that also from the same root we get the noun version of these words, in which (surprisingly) the ‑mit morphed into ‑mission. Thus: manumission, dismiss, mess and mission.