It's a phrasal verb—actually one of a few that can be used to roughly mean the same thing. These are both from Merriam-Webster:
informal
: to make a mistake : to do something incorrectly • About halfway into the recipe, I realized that I had messed up, and I had to start over. —often + on • She's afraid she'll mess up on the test. • I messed up on my first attempt.
informal
1 : to cause trouble for (someone) : to deal with (someone) in a way that may cause anger or violence • I wouldn't want to mess with him. • You'd better not be messing with me.
2 chiefly US : to deal with or be involved with (something that causes or that could cause trouble) • The company doesn't want to mess with small distributors. • He doesn't want to mess with cocaine anymore.
3 : to handle or play with (something) in a careless way : to mess around with (something) • Don't mess with the camera.
Also, a variation in slang is something like:
I'm gonna mess you up!
Which effectively means, I'm going to hurt you. (But it also has the implication of causing damage to your body so that various parts no longer function properly.)
The phrases used in your question are along the same lines as this—where somebody is causing deliberate harm to somebody or something.
It's quite idiomatic and understood informally. (Although it's not something that would normally be used in formal speech or writing.)
The Online Etymology Dictionary says this about the verb muck (bold text mine):
late 14c., "to dig in the ground," also "to remove manure," early 15c., "to spread manure, cover with muck," from muck (n.). Meaning "to make dirty" is from 1832; in the figurative sense, "to make a mess of," it is from 1886; to muck about "mess around" is from 1856. Related: Mucked; mucking.
As for the presence or absence of the preposition from, including it seems to be more common but there's no reason why it must be present. (And in those variations where an explicit subject is not present, its presence is actually wrong.)
✔ I'm going to stop them messing with my day.
✔ I'm going to stop them from messing with my day.
✔ Stop messing me up!
✘ Stop from messing me up!
✔ Stop yourself from messing up.
✘ Stop from messing up.
✔ Stop messing up.
There is specific discussion of the use of with in the question "Do you “prevent somebody doing something”, or "prevent somebody [from] doing something?" Note that it's mentioned there that it's more common (relatively speaking) to drop with in UK English than it is in US English.