To ‘stick (or put) your oar in’ means to interfere or try to have your say in somebody else’s life. It means to offer unwanted help or opinions to somebody who really wishes you would ‘keep your nose out’.
Imagine if you were happily rowing along in a rowing boat, pulling both oars yourself, but then your passenger produced their own oar and stuck it into the water to try to help take the boat in a different direction.
Rob: My Uncle Mark keeps trying to tell me which universities to apply for. It’s so annoying! I wish he wouldn’t keep sticking his oar in.