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.