It would be easy to think this is just a statement of agricultural fact, but remember,
All Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness -2 Timothy 3:16
So we need to be careful not to dismiss this as a mere statement about life on the farm; this is here in Scripture for a reason: to teach us and train us in how to be righteous. So the question is, why did God provide in Scripture a verse describing the way things are with oxen?
There is another verse about oxen that is explained more explicitly in Scripture:
You shall not muzzle the ox while he is threshing. -Deuteronomy 25:4
Paul explains the meaning as follows:
God is not concerned about oxen, is He? Or is He speaking altogether for our sake? Yes, for our sake it was written, because the plowman ought to plow in hope, and the thresher to thresh in hope of sharing the crops. If we sowed spiritual things in you, is it too much if we reap material things from you? -1 Corinthians 9:9-11
Paul is clarifying that this Old Testament law about oxen was intended to illustrate to God's people that a minister should not be "muzzled" while he is ministering. Just as it would be cruel to cover an ox's mouth while he is threshing in your field, it is cruel to deprive a minister of support while he is ministering to you.
Some teach that the verse in Deuteronomy provided an agricultural law which teaches a principle that applies to everything in life, whether it be oxen, ministers, or others. However, Paul clarifies that God was not concerned about oxen when He gave this instruction, but that it was given for the sake of those in ministry.
Here is an alternate reading of Proverbs 14:4
Where no oxen are, the manger is clean, but much revenue comes by the strength of the ox. -NASB
I have always taken this as follows:
Oxen are dirty, messy, cumbersome beasts, and yes, you can keep your manger a lot cleaner if you don't keep any oxen there... but, oxen are actually very profitable due to their great strength.
I had a Pastor tell me one time that I was like a "bull in a china shop" in church. (In other words, a big, troublesome beast who makes a mess of an otherwise pretty place.) I have always read this passage and thought "yeah, but God still uses me powerfully, so it's a good thing my church keeps me around."
I think this verse was given "for our sake", and it makes a lot of sense in church. The servants of God are like oxen. We can sure make a mess of things and be a headache to manage, and things are a lot simpler when we're not around... but, we are very profitable to the Kingdom, so it's good to keep us around, regardless of how difficult we are at times.
Anyway, that's just my take. I would suggest praying about it and asking God to give you the wisdom to know whether this is correct.